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cam is a farmer . each year he plants seeds that turn into corn . the function below gives the amount of corn , $ c $ , in kilograms ( kg ) , that he expects to produce if he plants corn on $ a $ acres of land . $ c ( a ) = 7500a - 1500 $ for example , if cam plants two , he expects to produce $ c ( 2 ) = 7500 ( 2 ) -1500= { 13 { , } { 500 } } $ $ \text { kg } $ of corn . what cam really wants to know is how much money he will make from selling this corn . so he uses the following function to predict the amount of money , $ m $ , in dollars , that he will earn from selling $ c $ kilograms of corn . $ m ( c ) = 0.9c - 50 $ so if cam produces 13.500 kg of corn , he can expect to make $ m ( { 13 { , } 500 } ) =0.9 ( { 13 { , } 500 } ) -50=\ $ { 12 { , } 100 } $ . notice that cam has to use two separate functions to get from acres planted to expected earnings . the first function , $ c $ , takes acres to corn , while the second function , $ m $ , takes corn to money . would n't it be great if cam could write a function that turned planted acres directly into expected earnings ? creating a new function we can indeed find the function that takes acres planted directly to expected earnings ! to find this new function , let 's think about the most general question : how much money does cam expect to make if he plants corn seed on $ a $ acres of land ? well , if cam plants corn on $ a $ acres , he expects to produce $ c ( a ) $ kilograms of corn . and if he produces $ c ( a ) $ kilograms of corn , he expects to makes $ m ( c ( a ) ) $ dollars . so , to find a general rule that converts $ a $ acres directly into expected earnings , we can find the expression $ m ( c ( a ) ) $ . but just how do we do this ? well , notice that in the expression $ m ( \greend { c ( a ) } ) $ , the input of function $ m $ is $ \greend { c ( a ) } $ . so , to find this expression , we can substitute $ \greend { c ( a ) } $ in for $ \goldd { c } $ in function $ m $ . $ \begin { align } m ( \goldd c ) & amp ; =0.9\goldd c-50\\ m ( { \greend { c ( a ) } } ) & amp ; =0.9 ( \greend { c ( a ) } ) -50\ \ & amp ; = 0.9 ( \greend { 7500a-1500 } ) -50~~~~~~~~~~\small { \gray { \text { since } } } \small { \gray { c ( a ) =7500a-1500 } } \\ & amp ; = 6750a-1350-50\\ & amp ; =6750a-1400 \end { align } $ so the function $ m ( c ( a ) ) =6750a-1400 $ converts acres planted directly into expected earnings . let 's use this new function to predict the amount of money that cam would make from planting corn on two acres . $ m ( c ( 2 ) ) =6750 ( 2 ) -1400=\ $ 12 { , } 100 $ cam can expect to make $ \ $ 12 { , } 100 $ from planting corn on two acres of land , which is consistent with our previous work ! defining composite functions we just found what is called a composite function . instead of substituting acres planted into the corn function , and then substituting the amount of corn produced into the money function , we found a function that takes the acres planted directly to the expected earnings . we did this by substituting $ c ( a ) $ into function $ m $ , or by finding $ m ( c ( a ) ) $ . let 's call this new function $ m\circ c $ , which is read as `` $ m $ composed with $ c $ '' . we now know that $ ( m\circ c ) ( a ) =m ( c ( a ) ) $ . this , in fact , is the formal definition of function composition ! visualizing the two methods here 's a visual to help interpret the above definition . using both functions $ c $ and $ m $ , function $ c $ —the corn function—takes two to 13,500 . then , function $ m $ —the money function—takes 13,500 to $ \ $ $ 12,100 . using the composite function , we see that function $ m\circ c $ takes two directly to $ \ $ $ 12,100 . the two are equivalent ! now let 's practice some problems . problem 2 ben is a potato farmer . the function $ p ( a ) = 25 { , } 000a - 1000 $ gives the amount of potatoes , $ p $ , in kilograms , that he expects to produce from planting potatoes on $ a $ acres of land . the function $ m ( p ) = 0.2p - 200 $ gives the amount of money , $ m $ , in dollars , that ben expects to make if he produces $ p $ kilograms of potatoes . problem 3
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now let 's practice some problems . problem 2 ben is a potato farmer . the function $ p ( a ) = 25 { , } 000a - 1000 $ gives the amount of potatoes , $ p $ , in kilograms , that he expects to produce from planting potatoes on $ a $ acres of land .
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f ( x ) = 2x-1 g ( x ) =3x and h ( x ) =x^2+1 how would i find the answer f ( g ( h ( 2 ) ) ) ?
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cam is a farmer . each year he plants seeds that turn into corn . the function below gives the amount of corn , $ c $ , in kilograms ( kg ) , that he expects to produce if he plants corn on $ a $ acres of land . $ c ( a ) = 7500a - 1500 $ for example , if cam plants two , he expects to produce $ c ( 2 ) = 7500 ( 2 ) -1500= { 13 { , } { 500 } } $ $ \text { kg } $ of corn . what cam really wants to know is how much money he will make from selling this corn . so he uses the following function to predict the amount of money , $ m $ , in dollars , that he will earn from selling $ c $ kilograms of corn . $ m ( c ) = 0.9c - 50 $ so if cam produces 13.500 kg of corn , he can expect to make $ m ( { 13 { , } 500 } ) =0.9 ( { 13 { , } 500 } ) -50=\ $ { 12 { , } 100 } $ . notice that cam has to use two separate functions to get from acres planted to expected earnings . the first function , $ c $ , takes acres to corn , while the second function , $ m $ , takes corn to money . would n't it be great if cam could write a function that turned planted acres directly into expected earnings ? creating a new function we can indeed find the function that takes acres planted directly to expected earnings ! to find this new function , let 's think about the most general question : how much money does cam expect to make if he plants corn seed on $ a $ acres of land ? well , if cam plants corn on $ a $ acres , he expects to produce $ c ( a ) $ kilograms of corn . and if he produces $ c ( a ) $ kilograms of corn , he expects to makes $ m ( c ( a ) ) $ dollars . so , to find a general rule that converts $ a $ acres directly into expected earnings , we can find the expression $ m ( c ( a ) ) $ . but just how do we do this ? well , notice that in the expression $ m ( \greend { c ( a ) } ) $ , the input of function $ m $ is $ \greend { c ( a ) } $ . so , to find this expression , we can substitute $ \greend { c ( a ) } $ in for $ \goldd { c } $ in function $ m $ . $ \begin { align } m ( \goldd c ) & amp ; =0.9\goldd c-50\\ m ( { \greend { c ( a ) } } ) & amp ; =0.9 ( \greend { c ( a ) } ) -50\ \ & amp ; = 0.9 ( \greend { 7500a-1500 } ) -50~~~~~~~~~~\small { \gray { \text { since } } } \small { \gray { c ( a ) =7500a-1500 } } \\ & amp ; = 6750a-1350-50\\ & amp ; =6750a-1400 \end { align } $ so the function $ m ( c ( a ) ) =6750a-1400 $ converts acres planted directly into expected earnings . let 's use this new function to predict the amount of money that cam would make from planting corn on two acres . $ m ( c ( 2 ) ) =6750 ( 2 ) -1400=\ $ 12 { , } 100 $ cam can expect to make $ \ $ 12 { , } 100 $ from planting corn on two acres of land , which is consistent with our previous work ! defining composite functions we just found what is called a composite function . instead of substituting acres planted into the corn function , and then substituting the amount of corn produced into the money function , we found a function that takes the acres planted directly to the expected earnings . we did this by substituting $ c ( a ) $ into function $ m $ , or by finding $ m ( c ( a ) ) $ . let 's call this new function $ m\circ c $ , which is read as `` $ m $ composed with $ c $ '' . we now know that $ ( m\circ c ) ( a ) =m ( c ( a ) ) $ . this , in fact , is the formal definition of function composition ! visualizing the two methods here 's a visual to help interpret the above definition . using both functions $ c $ and $ m $ , function $ c $ —the corn function—takes two to 13,500 . then , function $ m $ —the money function—takes 13,500 to $ \ $ $ 12,100 . using the composite function , we see that function $ m\circ c $ takes two directly to $ \ $ $ 12,100 . the two are equivalent ! now let 's practice some problems . problem 2 ben is a potato farmer . the function $ p ( a ) = 25 { , } 000a - 1000 $ gives the amount of potatoes , $ p $ , in kilograms , that he expects to produce from planting potatoes on $ a $ acres of land . the function $ m ( p ) = 0.2p - 200 $ gives the amount of money , $ m $ , in dollars , that ben expects to make if he produces $ p $ kilograms of potatoes . problem 3
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so , to find this expression , we can substitute $ \greend { c ( a ) } $ in for $ \goldd { c } $ in function $ m $ . $ \begin { align } m ( \goldd c ) & amp ; =0.9\goldd c-50\\ m ( { \greend { c ( a ) } } ) & amp ; =0.9 ( \greend { c ( a ) } ) -50\ \ & amp ; = 0.9 ( \greend { 7500a-1500 } ) -50~~~~~~~~~~\small { \gray { \text { since } } } \small { \gray { c ( a ) =7500a-1500 } } \\ & amp ; = 6750a-1350-50\\ & amp ; =6750a-1400 \end { align } $ so the function $ m ( c ( a ) ) =6750a-1400 $ converts acres planted directly into expected earnings . let 's use this new function to predict the amount of money that cam would make from planting corn on two acres .
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how does 6750a-1350-50 turn into 6750a-1400 instead of 6750a-1300 ?
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cam is a farmer . each year he plants seeds that turn into corn . the function below gives the amount of corn , $ c $ , in kilograms ( kg ) , that he expects to produce if he plants corn on $ a $ acres of land . $ c ( a ) = 7500a - 1500 $ for example , if cam plants two , he expects to produce $ c ( 2 ) = 7500 ( 2 ) -1500= { 13 { , } { 500 } } $ $ \text { kg } $ of corn . what cam really wants to know is how much money he will make from selling this corn . so he uses the following function to predict the amount of money , $ m $ , in dollars , that he will earn from selling $ c $ kilograms of corn . $ m ( c ) = 0.9c - 50 $ so if cam produces 13.500 kg of corn , he can expect to make $ m ( { 13 { , } 500 } ) =0.9 ( { 13 { , } 500 } ) -50=\ $ { 12 { , } 100 } $ . notice that cam has to use two separate functions to get from acres planted to expected earnings . the first function , $ c $ , takes acres to corn , while the second function , $ m $ , takes corn to money . would n't it be great if cam could write a function that turned planted acres directly into expected earnings ? creating a new function we can indeed find the function that takes acres planted directly to expected earnings ! to find this new function , let 's think about the most general question : how much money does cam expect to make if he plants corn seed on $ a $ acres of land ? well , if cam plants corn on $ a $ acres , he expects to produce $ c ( a ) $ kilograms of corn . and if he produces $ c ( a ) $ kilograms of corn , he expects to makes $ m ( c ( a ) ) $ dollars . so , to find a general rule that converts $ a $ acres directly into expected earnings , we can find the expression $ m ( c ( a ) ) $ . but just how do we do this ? well , notice that in the expression $ m ( \greend { c ( a ) } ) $ , the input of function $ m $ is $ \greend { c ( a ) } $ . so , to find this expression , we can substitute $ \greend { c ( a ) } $ in for $ \goldd { c } $ in function $ m $ . $ \begin { align } m ( \goldd c ) & amp ; =0.9\goldd c-50\\ m ( { \greend { c ( a ) } } ) & amp ; =0.9 ( \greend { c ( a ) } ) -50\ \ & amp ; = 0.9 ( \greend { 7500a-1500 } ) -50~~~~~~~~~~\small { \gray { \text { since } } } \small { \gray { c ( a ) =7500a-1500 } } \\ & amp ; = 6750a-1350-50\\ & amp ; =6750a-1400 \end { align } $ so the function $ m ( c ( a ) ) =6750a-1400 $ converts acres planted directly into expected earnings . let 's use this new function to predict the amount of money that cam would make from planting corn on two acres . $ m ( c ( 2 ) ) =6750 ( 2 ) -1400=\ $ 12 { , } 100 $ cam can expect to make $ \ $ 12 { , } 100 $ from planting corn on two acres of land , which is consistent with our previous work ! defining composite functions we just found what is called a composite function . instead of substituting acres planted into the corn function , and then substituting the amount of corn produced into the money function , we found a function that takes the acres planted directly to the expected earnings . we did this by substituting $ c ( a ) $ into function $ m $ , or by finding $ m ( c ( a ) ) $ . let 's call this new function $ m\circ c $ , which is read as `` $ m $ composed with $ c $ '' . we now know that $ ( m\circ c ) ( a ) =m ( c ( a ) ) $ . this , in fact , is the formal definition of function composition ! visualizing the two methods here 's a visual to help interpret the above definition . using both functions $ c $ and $ m $ , function $ c $ —the corn function—takes two to 13,500 . then , function $ m $ —the money function—takes 13,500 to $ \ $ $ 12,100 . using the composite function , we see that function $ m\circ c $ takes two directly to $ \ $ $ 12,100 . the two are equivalent ! now let 's practice some problems . problem 2 ben is a potato farmer . the function $ p ( a ) = 25 { , } 000a - 1000 $ gives the amount of potatoes , $ p $ , in kilograms , that he expects to produce from planting potatoes on $ a $ acres of land . the function $ m ( p ) = 0.2p - 200 $ gives the amount of money , $ m $ , in dollars , that ben expects to make if he produces $ p $ kilograms of potatoes . problem 3
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visualizing the two methods here 's a visual to help interpret the above definition . using both functions $ c $ and $ m $ , function $ c $ —the corn function—takes two to 13,500 . then , function $ m $ —the money function—takes 13,500 to $ \ $ $ 12,100 . using the composite function , we see that function $ m\circ c $ takes two directly to $ \ $ $ 12,100 .
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what is the difference between a regular function and a composing function ?
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cam is a farmer . each year he plants seeds that turn into corn . the function below gives the amount of corn , $ c $ , in kilograms ( kg ) , that he expects to produce if he plants corn on $ a $ acres of land . $ c ( a ) = 7500a - 1500 $ for example , if cam plants two , he expects to produce $ c ( 2 ) = 7500 ( 2 ) -1500= { 13 { , } { 500 } } $ $ \text { kg } $ of corn . what cam really wants to know is how much money he will make from selling this corn . so he uses the following function to predict the amount of money , $ m $ , in dollars , that he will earn from selling $ c $ kilograms of corn . $ m ( c ) = 0.9c - 50 $ so if cam produces 13.500 kg of corn , he can expect to make $ m ( { 13 { , } 500 } ) =0.9 ( { 13 { , } 500 } ) -50=\ $ { 12 { , } 100 } $ . notice that cam has to use two separate functions to get from acres planted to expected earnings . the first function , $ c $ , takes acres to corn , while the second function , $ m $ , takes corn to money . would n't it be great if cam could write a function that turned planted acres directly into expected earnings ? creating a new function we can indeed find the function that takes acres planted directly to expected earnings ! to find this new function , let 's think about the most general question : how much money does cam expect to make if he plants corn seed on $ a $ acres of land ? well , if cam plants corn on $ a $ acres , he expects to produce $ c ( a ) $ kilograms of corn . and if he produces $ c ( a ) $ kilograms of corn , he expects to makes $ m ( c ( a ) ) $ dollars . so , to find a general rule that converts $ a $ acres directly into expected earnings , we can find the expression $ m ( c ( a ) ) $ . but just how do we do this ? well , notice that in the expression $ m ( \greend { c ( a ) } ) $ , the input of function $ m $ is $ \greend { c ( a ) } $ . so , to find this expression , we can substitute $ \greend { c ( a ) } $ in for $ \goldd { c } $ in function $ m $ . $ \begin { align } m ( \goldd c ) & amp ; =0.9\goldd c-50\\ m ( { \greend { c ( a ) } } ) & amp ; =0.9 ( \greend { c ( a ) } ) -50\ \ & amp ; = 0.9 ( \greend { 7500a-1500 } ) -50~~~~~~~~~~\small { \gray { \text { since } } } \small { \gray { c ( a ) =7500a-1500 } } \\ & amp ; = 6750a-1350-50\\ & amp ; =6750a-1400 \end { align } $ so the function $ m ( c ( a ) ) =6750a-1400 $ converts acres planted directly into expected earnings . let 's use this new function to predict the amount of money that cam would make from planting corn on two acres . $ m ( c ( 2 ) ) =6750 ( 2 ) -1400=\ $ 12 { , } 100 $ cam can expect to make $ \ $ 12 { , } 100 $ from planting corn on two acres of land , which is consistent with our previous work ! defining composite functions we just found what is called a composite function . instead of substituting acres planted into the corn function , and then substituting the amount of corn produced into the money function , we found a function that takes the acres planted directly to the expected earnings . we did this by substituting $ c ( a ) $ into function $ m $ , or by finding $ m ( c ( a ) ) $ . let 's call this new function $ m\circ c $ , which is read as `` $ m $ composed with $ c $ '' . we now know that $ ( m\circ c ) ( a ) =m ( c ( a ) ) $ . this , in fact , is the formal definition of function composition ! visualizing the two methods here 's a visual to help interpret the above definition . using both functions $ c $ and $ m $ , function $ c $ —the corn function—takes two to 13,500 . then , function $ m $ —the money function—takes 13,500 to $ \ $ $ 12,100 . using the composite function , we see that function $ m\circ c $ takes two directly to $ \ $ $ 12,100 . the two are equivalent ! now let 's practice some problems . problem 2 ben is a potato farmer . the function $ p ( a ) = 25 { , } 000a - 1000 $ gives the amount of potatoes , $ p $ , in kilograms , that he expects to produce from planting potatoes on $ a $ acres of land . the function $ m ( p ) = 0.2p - 200 $ gives the amount of money , $ m $ , in dollars , that ben expects to make if he produces $ p $ kilograms of potatoes . problem 3
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creating a new function we can indeed find the function that takes acres planted directly to expected earnings ! to find this new function , let 's think about the most general question : how much money does cam expect to make if he plants corn seed on $ a $ acres of land ? well , if cam plants corn on $ a $ acres , he expects to produce $ c ( a ) $ kilograms of corn .
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how much money can ben expect to make if he sells all of the potatoes produced on the 3 acres ?
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cam is a farmer . each year he plants seeds that turn into corn . the function below gives the amount of corn , $ c $ , in kilograms ( kg ) , that he expects to produce if he plants corn on $ a $ acres of land . $ c ( a ) = 7500a - 1500 $ for example , if cam plants two , he expects to produce $ c ( 2 ) = 7500 ( 2 ) -1500= { 13 { , } { 500 } } $ $ \text { kg } $ of corn . what cam really wants to know is how much money he will make from selling this corn . so he uses the following function to predict the amount of money , $ m $ , in dollars , that he will earn from selling $ c $ kilograms of corn . $ m ( c ) = 0.9c - 50 $ so if cam produces 13.500 kg of corn , he can expect to make $ m ( { 13 { , } 500 } ) =0.9 ( { 13 { , } 500 } ) -50=\ $ { 12 { , } 100 } $ . notice that cam has to use two separate functions to get from acres planted to expected earnings . the first function , $ c $ , takes acres to corn , while the second function , $ m $ , takes corn to money . would n't it be great if cam could write a function that turned planted acres directly into expected earnings ? creating a new function we can indeed find the function that takes acres planted directly to expected earnings ! to find this new function , let 's think about the most general question : how much money does cam expect to make if he plants corn seed on $ a $ acres of land ? well , if cam plants corn on $ a $ acres , he expects to produce $ c ( a ) $ kilograms of corn . and if he produces $ c ( a ) $ kilograms of corn , he expects to makes $ m ( c ( a ) ) $ dollars . so , to find a general rule that converts $ a $ acres directly into expected earnings , we can find the expression $ m ( c ( a ) ) $ . but just how do we do this ? well , notice that in the expression $ m ( \greend { c ( a ) } ) $ , the input of function $ m $ is $ \greend { c ( a ) } $ . so , to find this expression , we can substitute $ \greend { c ( a ) } $ in for $ \goldd { c } $ in function $ m $ . $ \begin { align } m ( \goldd c ) & amp ; =0.9\goldd c-50\\ m ( { \greend { c ( a ) } } ) & amp ; =0.9 ( \greend { c ( a ) } ) -50\ \ & amp ; = 0.9 ( \greend { 7500a-1500 } ) -50~~~~~~~~~~\small { \gray { \text { since } } } \small { \gray { c ( a ) =7500a-1500 } } \\ & amp ; = 6750a-1350-50\\ & amp ; =6750a-1400 \end { align } $ so the function $ m ( c ( a ) ) =6750a-1400 $ converts acres planted directly into expected earnings . let 's use this new function to predict the amount of money that cam would make from planting corn on two acres . $ m ( c ( 2 ) ) =6750 ( 2 ) -1400=\ $ 12 { , } 100 $ cam can expect to make $ \ $ 12 { , } 100 $ from planting corn on two acres of land , which is consistent with our previous work ! defining composite functions we just found what is called a composite function . instead of substituting acres planted into the corn function , and then substituting the amount of corn produced into the money function , we found a function that takes the acres planted directly to the expected earnings . we did this by substituting $ c ( a ) $ into function $ m $ , or by finding $ m ( c ( a ) ) $ . let 's call this new function $ m\circ c $ , which is read as `` $ m $ composed with $ c $ '' . we now know that $ ( m\circ c ) ( a ) =m ( c ( a ) ) $ . this , in fact , is the formal definition of function composition ! visualizing the two methods here 's a visual to help interpret the above definition . using both functions $ c $ and $ m $ , function $ c $ —the corn function—takes two to 13,500 . then , function $ m $ —the money function—takes 13,500 to $ \ $ $ 12,100 . using the composite function , we see that function $ m\circ c $ takes two directly to $ \ $ $ 12,100 . the two are equivalent ! now let 's practice some problems . problem 2 ben is a potato farmer . the function $ p ( a ) = 25 { , } 000a - 1000 $ gives the amount of potatoes , $ p $ , in kilograms , that he expects to produce from planting potatoes on $ a $ acres of land . the function $ m ( p ) = 0.2p - 200 $ gives the amount of money , $ m $ , in dollars , that ben expects to make if he produces $ p $ kilograms of potatoes . problem 3
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$ m ( c ( 2 ) ) =6750 ( 2 ) -1400=\ $ 12 { , } 100 $ cam can expect to make $ \ $ 12 { , } 100 $ from planting corn on two acres of land , which is consistent with our previous work ! defining composite functions we just found what is called a composite function . instead of substituting acres planted into the corn function , and then substituting the amount of corn produced into the money function , we found a function that takes the acres planted directly to the expected earnings .
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would the reverse of these functions be true ?
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cam is a farmer . each year he plants seeds that turn into corn . the function below gives the amount of corn , $ c $ , in kilograms ( kg ) , that he expects to produce if he plants corn on $ a $ acres of land . $ c ( a ) = 7500a - 1500 $ for example , if cam plants two , he expects to produce $ c ( 2 ) = 7500 ( 2 ) -1500= { 13 { , } { 500 } } $ $ \text { kg } $ of corn . what cam really wants to know is how much money he will make from selling this corn . so he uses the following function to predict the amount of money , $ m $ , in dollars , that he will earn from selling $ c $ kilograms of corn . $ m ( c ) = 0.9c - 50 $ so if cam produces 13.500 kg of corn , he can expect to make $ m ( { 13 { , } 500 } ) =0.9 ( { 13 { , } 500 } ) -50=\ $ { 12 { , } 100 } $ . notice that cam has to use two separate functions to get from acres planted to expected earnings . the first function , $ c $ , takes acres to corn , while the second function , $ m $ , takes corn to money . would n't it be great if cam could write a function that turned planted acres directly into expected earnings ? creating a new function we can indeed find the function that takes acres planted directly to expected earnings ! to find this new function , let 's think about the most general question : how much money does cam expect to make if he plants corn seed on $ a $ acres of land ? well , if cam plants corn on $ a $ acres , he expects to produce $ c ( a ) $ kilograms of corn . and if he produces $ c ( a ) $ kilograms of corn , he expects to makes $ m ( c ( a ) ) $ dollars . so , to find a general rule that converts $ a $ acres directly into expected earnings , we can find the expression $ m ( c ( a ) ) $ . but just how do we do this ? well , notice that in the expression $ m ( \greend { c ( a ) } ) $ , the input of function $ m $ is $ \greend { c ( a ) } $ . so , to find this expression , we can substitute $ \greend { c ( a ) } $ in for $ \goldd { c } $ in function $ m $ . $ \begin { align } m ( \goldd c ) & amp ; =0.9\goldd c-50\\ m ( { \greend { c ( a ) } } ) & amp ; =0.9 ( \greend { c ( a ) } ) -50\ \ & amp ; = 0.9 ( \greend { 7500a-1500 } ) -50~~~~~~~~~~\small { \gray { \text { since } } } \small { \gray { c ( a ) =7500a-1500 } } \\ & amp ; = 6750a-1350-50\\ & amp ; =6750a-1400 \end { align } $ so the function $ m ( c ( a ) ) =6750a-1400 $ converts acres planted directly into expected earnings . let 's use this new function to predict the amount of money that cam would make from planting corn on two acres . $ m ( c ( 2 ) ) =6750 ( 2 ) -1400=\ $ 12 { , } 100 $ cam can expect to make $ \ $ 12 { , } 100 $ from planting corn on two acres of land , which is consistent with our previous work ! defining composite functions we just found what is called a composite function . instead of substituting acres planted into the corn function , and then substituting the amount of corn produced into the money function , we found a function that takes the acres planted directly to the expected earnings . we did this by substituting $ c ( a ) $ into function $ m $ , or by finding $ m ( c ( a ) ) $ . let 's call this new function $ m\circ c $ , which is read as `` $ m $ composed with $ c $ '' . we now know that $ ( m\circ c ) ( a ) =m ( c ( a ) ) $ . this , in fact , is the formal definition of function composition ! visualizing the two methods here 's a visual to help interpret the above definition . using both functions $ c $ and $ m $ , function $ c $ —the corn function—takes two to 13,500 . then , function $ m $ —the money function—takes 13,500 to $ \ $ $ 12,100 . using the composite function , we see that function $ m\circ c $ takes two directly to $ \ $ $ 12,100 . the two are equivalent ! now let 's practice some problems . problem 2 ben is a potato farmer . the function $ p ( a ) = 25 { , } 000a - 1000 $ gives the amount of potatoes , $ p $ , in kilograms , that he expects to produce from planting potatoes on $ a $ acres of land . the function $ m ( p ) = 0.2p - 200 $ gives the amount of money , $ m $ , in dollars , that ben expects to make if he produces $ p $ kilograms of potatoes . problem 3
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$ \begin { align } m ( \goldd c ) & amp ; =0.9\goldd c-50\\ m ( { \greend { c ( a ) } } ) & amp ; =0.9 ( \greend { c ( a ) } ) -50\ \ & amp ; = 0.9 ( \greend { 7500a-1500 } ) -50~~~~~~~~~~\small { \gray { \text { since } } } \small { \gray { c ( a ) =7500a-1500 } } \\ & amp ; = 6750a-1350-50\\ & amp ; =6750a-1400 \end { align } $ so the function $ m ( c ( a ) ) =6750a-1400 $ converts acres planted directly into expected earnings . let 's use this new function to predict the amount of money that cam would make from planting corn on two acres . $ m ( c ( 2 ) ) =6750 ( 2 ) -1400=\ $ 12 { , } 100 $ cam can expect to make $ \ $ 12 { , } 100 $ from planting corn on two acres of land , which is consistent with our previous work !
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if we wanted to find the area needed to plant enough corn to make an x amount of money be a ( c ( m ) ) ?
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cam is a farmer . each year he plants seeds that turn into corn . the function below gives the amount of corn , $ c $ , in kilograms ( kg ) , that he expects to produce if he plants corn on $ a $ acres of land . $ c ( a ) = 7500a - 1500 $ for example , if cam plants two , he expects to produce $ c ( 2 ) = 7500 ( 2 ) -1500= { 13 { , } { 500 } } $ $ \text { kg } $ of corn . what cam really wants to know is how much money he will make from selling this corn . so he uses the following function to predict the amount of money , $ m $ , in dollars , that he will earn from selling $ c $ kilograms of corn . $ m ( c ) = 0.9c - 50 $ so if cam produces 13.500 kg of corn , he can expect to make $ m ( { 13 { , } 500 } ) =0.9 ( { 13 { , } 500 } ) -50=\ $ { 12 { , } 100 } $ . notice that cam has to use two separate functions to get from acres planted to expected earnings . the first function , $ c $ , takes acres to corn , while the second function , $ m $ , takes corn to money . would n't it be great if cam could write a function that turned planted acres directly into expected earnings ? creating a new function we can indeed find the function that takes acres planted directly to expected earnings ! to find this new function , let 's think about the most general question : how much money does cam expect to make if he plants corn seed on $ a $ acres of land ? well , if cam plants corn on $ a $ acres , he expects to produce $ c ( a ) $ kilograms of corn . and if he produces $ c ( a ) $ kilograms of corn , he expects to makes $ m ( c ( a ) ) $ dollars . so , to find a general rule that converts $ a $ acres directly into expected earnings , we can find the expression $ m ( c ( a ) ) $ . but just how do we do this ? well , notice that in the expression $ m ( \greend { c ( a ) } ) $ , the input of function $ m $ is $ \greend { c ( a ) } $ . so , to find this expression , we can substitute $ \greend { c ( a ) } $ in for $ \goldd { c } $ in function $ m $ . $ \begin { align } m ( \goldd c ) & amp ; =0.9\goldd c-50\\ m ( { \greend { c ( a ) } } ) & amp ; =0.9 ( \greend { c ( a ) } ) -50\ \ & amp ; = 0.9 ( \greend { 7500a-1500 } ) -50~~~~~~~~~~\small { \gray { \text { since } } } \small { \gray { c ( a ) =7500a-1500 } } \\ & amp ; = 6750a-1350-50\\ & amp ; =6750a-1400 \end { align } $ so the function $ m ( c ( a ) ) =6750a-1400 $ converts acres planted directly into expected earnings . let 's use this new function to predict the amount of money that cam would make from planting corn on two acres . $ m ( c ( 2 ) ) =6750 ( 2 ) -1400=\ $ 12 { , } 100 $ cam can expect to make $ \ $ 12 { , } 100 $ from planting corn on two acres of land , which is consistent with our previous work ! defining composite functions we just found what is called a composite function . instead of substituting acres planted into the corn function , and then substituting the amount of corn produced into the money function , we found a function that takes the acres planted directly to the expected earnings . we did this by substituting $ c ( a ) $ into function $ m $ , or by finding $ m ( c ( a ) ) $ . let 's call this new function $ m\circ c $ , which is read as `` $ m $ composed with $ c $ '' . we now know that $ ( m\circ c ) ( a ) =m ( c ( a ) ) $ . this , in fact , is the formal definition of function composition ! visualizing the two methods here 's a visual to help interpret the above definition . using both functions $ c $ and $ m $ , function $ c $ —the corn function—takes two to 13,500 . then , function $ m $ —the money function—takes 13,500 to $ \ $ $ 12,100 . using the composite function , we see that function $ m\circ c $ takes two directly to $ \ $ $ 12,100 . the two are equivalent ! now let 's practice some problems . problem 2 ben is a potato farmer . the function $ p ( a ) = 25 { , } 000a - 1000 $ gives the amount of potatoes , $ p $ , in kilograms , that he expects to produce from planting potatoes on $ a $ acres of land . the function $ m ( p ) = 0.2p - 200 $ gives the amount of money , $ m $ , in dollars , that ben expects to make if he produces $ p $ kilograms of potatoes . problem 3
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$ \begin { align } m ( \goldd c ) & amp ; =0.9\goldd c-50\\ m ( { \greend { c ( a ) } } ) & amp ; =0.9 ( \greend { c ( a ) } ) -50\ \ & amp ; = 0.9 ( \greend { 7500a-1500 } ) -50~~~~~~~~~~\small { \gray { \text { since } } } \small { \gray { c ( a ) =7500a-1500 } } \\ & amp ; = 6750a-1350-50\\ & amp ; =6750a-1400 \end { align } $ so the function $ m ( c ( a ) ) =6750a-1400 $ converts acres planted directly into expected earnings . let 's use this new function to predict the amount of money that cam would make from planting corn on two acres . $ m ( c ( 2 ) ) =6750 ( 2 ) -1400=\ $ 12 { , } 100 $ cam can expect to make $ \ $ 12 { , } 100 $ from planting corn on two acres of land , which is consistent with our previous work !
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would we need to create new functions ?
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cam is a farmer . each year he plants seeds that turn into corn . the function below gives the amount of corn , $ c $ , in kilograms ( kg ) , that he expects to produce if he plants corn on $ a $ acres of land . $ c ( a ) = 7500a - 1500 $ for example , if cam plants two , he expects to produce $ c ( 2 ) = 7500 ( 2 ) -1500= { 13 { , } { 500 } } $ $ \text { kg } $ of corn . what cam really wants to know is how much money he will make from selling this corn . so he uses the following function to predict the amount of money , $ m $ , in dollars , that he will earn from selling $ c $ kilograms of corn . $ m ( c ) = 0.9c - 50 $ so if cam produces 13.500 kg of corn , he can expect to make $ m ( { 13 { , } 500 } ) =0.9 ( { 13 { , } 500 } ) -50=\ $ { 12 { , } 100 } $ . notice that cam has to use two separate functions to get from acres planted to expected earnings . the first function , $ c $ , takes acres to corn , while the second function , $ m $ , takes corn to money . would n't it be great if cam could write a function that turned planted acres directly into expected earnings ? creating a new function we can indeed find the function that takes acres planted directly to expected earnings ! to find this new function , let 's think about the most general question : how much money does cam expect to make if he plants corn seed on $ a $ acres of land ? well , if cam plants corn on $ a $ acres , he expects to produce $ c ( a ) $ kilograms of corn . and if he produces $ c ( a ) $ kilograms of corn , he expects to makes $ m ( c ( a ) ) $ dollars . so , to find a general rule that converts $ a $ acres directly into expected earnings , we can find the expression $ m ( c ( a ) ) $ . but just how do we do this ? well , notice that in the expression $ m ( \greend { c ( a ) } ) $ , the input of function $ m $ is $ \greend { c ( a ) } $ . so , to find this expression , we can substitute $ \greend { c ( a ) } $ in for $ \goldd { c } $ in function $ m $ . $ \begin { align } m ( \goldd c ) & amp ; =0.9\goldd c-50\\ m ( { \greend { c ( a ) } } ) & amp ; =0.9 ( \greend { c ( a ) } ) -50\ \ & amp ; = 0.9 ( \greend { 7500a-1500 } ) -50~~~~~~~~~~\small { \gray { \text { since } } } \small { \gray { c ( a ) =7500a-1500 } } \\ & amp ; = 6750a-1350-50\\ & amp ; =6750a-1400 \end { align } $ so the function $ m ( c ( a ) ) =6750a-1400 $ converts acres planted directly into expected earnings . let 's use this new function to predict the amount of money that cam would make from planting corn on two acres . $ m ( c ( 2 ) ) =6750 ( 2 ) -1400=\ $ 12 { , } 100 $ cam can expect to make $ \ $ 12 { , } 100 $ from planting corn on two acres of land , which is consistent with our previous work ! defining composite functions we just found what is called a composite function . instead of substituting acres planted into the corn function , and then substituting the amount of corn produced into the money function , we found a function that takes the acres planted directly to the expected earnings . we did this by substituting $ c ( a ) $ into function $ m $ , or by finding $ m ( c ( a ) ) $ . let 's call this new function $ m\circ c $ , which is read as `` $ m $ composed with $ c $ '' . we now know that $ ( m\circ c ) ( a ) =m ( c ( a ) ) $ . this , in fact , is the formal definition of function composition ! visualizing the two methods here 's a visual to help interpret the above definition . using both functions $ c $ and $ m $ , function $ c $ —the corn function—takes two to 13,500 . then , function $ m $ —the money function—takes 13,500 to $ \ $ $ 12,100 . using the composite function , we see that function $ m\circ c $ takes two directly to $ \ $ $ 12,100 . the two are equivalent ! now let 's practice some problems . problem 2 ben is a potato farmer . the function $ p ( a ) = 25 { , } 000a - 1000 $ gives the amount of potatoes , $ p $ , in kilograms , that he expects to produce from planting potatoes on $ a $ acres of land . the function $ m ( p ) = 0.2p - 200 $ gives the amount of money , $ m $ , in dollars , that ben expects to make if he produces $ p $ kilograms of potatoes . problem 3
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now let 's practice some problems . problem 2 ben is a potato farmer . the function $ p ( a ) = 25 { , } 000a - 1000 $ gives the amount of potatoes , $ p $ , in kilograms , that he expects to produce from planting potatoes on $ a $ acres of land .
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what would be the difference between ( f o g ) ( 2 ) and ( g o f ) ( 2 ) ?
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cam is a farmer . each year he plants seeds that turn into corn . the function below gives the amount of corn , $ c $ , in kilograms ( kg ) , that he expects to produce if he plants corn on $ a $ acres of land . $ c ( a ) = 7500a - 1500 $ for example , if cam plants two , he expects to produce $ c ( 2 ) = 7500 ( 2 ) -1500= { 13 { , } { 500 } } $ $ \text { kg } $ of corn . what cam really wants to know is how much money he will make from selling this corn . so he uses the following function to predict the amount of money , $ m $ , in dollars , that he will earn from selling $ c $ kilograms of corn . $ m ( c ) = 0.9c - 50 $ so if cam produces 13.500 kg of corn , he can expect to make $ m ( { 13 { , } 500 } ) =0.9 ( { 13 { , } 500 } ) -50=\ $ { 12 { , } 100 } $ . notice that cam has to use two separate functions to get from acres planted to expected earnings . the first function , $ c $ , takes acres to corn , while the second function , $ m $ , takes corn to money . would n't it be great if cam could write a function that turned planted acres directly into expected earnings ? creating a new function we can indeed find the function that takes acres planted directly to expected earnings ! to find this new function , let 's think about the most general question : how much money does cam expect to make if he plants corn seed on $ a $ acres of land ? well , if cam plants corn on $ a $ acres , he expects to produce $ c ( a ) $ kilograms of corn . and if he produces $ c ( a ) $ kilograms of corn , he expects to makes $ m ( c ( a ) ) $ dollars . so , to find a general rule that converts $ a $ acres directly into expected earnings , we can find the expression $ m ( c ( a ) ) $ . but just how do we do this ? well , notice that in the expression $ m ( \greend { c ( a ) } ) $ , the input of function $ m $ is $ \greend { c ( a ) } $ . so , to find this expression , we can substitute $ \greend { c ( a ) } $ in for $ \goldd { c } $ in function $ m $ . $ \begin { align } m ( \goldd c ) & amp ; =0.9\goldd c-50\\ m ( { \greend { c ( a ) } } ) & amp ; =0.9 ( \greend { c ( a ) } ) -50\ \ & amp ; = 0.9 ( \greend { 7500a-1500 } ) -50~~~~~~~~~~\small { \gray { \text { since } } } \small { \gray { c ( a ) =7500a-1500 } } \\ & amp ; = 6750a-1350-50\\ & amp ; =6750a-1400 \end { align } $ so the function $ m ( c ( a ) ) =6750a-1400 $ converts acres planted directly into expected earnings . let 's use this new function to predict the amount of money that cam would make from planting corn on two acres . $ m ( c ( 2 ) ) =6750 ( 2 ) -1400=\ $ 12 { , } 100 $ cam can expect to make $ \ $ 12 { , } 100 $ from planting corn on two acres of land , which is consistent with our previous work ! defining composite functions we just found what is called a composite function . instead of substituting acres planted into the corn function , and then substituting the amount of corn produced into the money function , we found a function that takes the acres planted directly to the expected earnings . we did this by substituting $ c ( a ) $ into function $ m $ , or by finding $ m ( c ( a ) ) $ . let 's call this new function $ m\circ c $ , which is read as `` $ m $ composed with $ c $ '' . we now know that $ ( m\circ c ) ( a ) =m ( c ( a ) ) $ . this , in fact , is the formal definition of function composition ! visualizing the two methods here 's a visual to help interpret the above definition . using both functions $ c $ and $ m $ , function $ c $ —the corn function—takes two to 13,500 . then , function $ m $ —the money function—takes 13,500 to $ \ $ $ 12,100 . using the composite function , we see that function $ m\circ c $ takes two directly to $ \ $ $ 12,100 . the two are equivalent ! now let 's practice some problems . problem 2 ben is a potato farmer . the function $ p ( a ) = 25 { , } 000a - 1000 $ gives the amount of potatoes , $ p $ , in kilograms , that he expects to produce from planting potatoes on $ a $ acres of land . the function $ m ( p ) = 0.2p - 200 $ gives the amount of money , $ m $ , in dollars , that ben expects to make if he produces $ p $ kilograms of potatoes . problem 3
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$ m ( c ( 2 ) ) =6750 ( 2 ) -1400=\ $ 12 { , } 100 $ cam can expect to make $ \ $ 12 { , } 100 $ from planting corn on two acres of land , which is consistent with our previous work ! defining composite functions we just found what is called a composite function . instead of substituting acres planted into the corn function , and then substituting the amount of corn produced into the money function , we found a function that takes the acres planted directly to the expected earnings .
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my question : when you speak of functions such as f ( x ) or g ( x ) and then define them as you do on the videos as , for example f ( x ) =9+x2 or g ( x ) +5x2 + 2x +1 , where do these equations come from ?
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this list links to videos , essays , images , and additional resources for the 250 required works of art for the ap* art history course and exam . smarthistory is adding new materials regularly . for more context on these works of art , please see the art history sections of khan academy . content area 1 global prehistory 30,000-500b.c.e . 1 . apollo 11 stones ( essay , image , additional resources ) 2 . great hall of bulls ( essay , image , additional resources ) 3 . camelid sacrum in the shape of a canine ( essay , image , additional resources ) 4 . running horned woman ( essay , image , additional resources ) 5 . bushel with ibex motifs ( video , images , additional resources ) 6 . anthropomorphic stele ( essay , image , additional resources ) 7 . jade cong ( video , article , image , additional resources ) 8 . stonehenge ( essay , video , image 1 , image 2 , additional resources ) 9 . ambum stone ( essay , image , additional resources ) 10 . tlatilco female figurine ( video , essay , image , additional resources ) 11 . terracotta fragment ( lapita ) ( essay , image , additional resources ) content area 2 ancient mediterranean 3,500-300 b.c.e . 12 . white temple and its ziggurat ( essay , image 1 , image 2 additional resources ) 13 . palette of king narmer ( essay , images , additional resources ) 14 . statues of votive figures , from the square temple at eshnunna ( video , images , additional resources ) 15 . seated scribe ( video , images , additional resources ) 16 . standard of ur from the royal tombs at ur ( video , essay , images , additional resources 17 . great pyramids of giza ( essay , images , additional resources ) a. pyramid of khufu ( essay , additional resources ) b. pyramid of khafre and the great sphinx ( essay , additional resources ) c. pyramid of menkaura ( essay , additional resources ) 18 . king menkaura and queen ( essay , images , additional resources ) 19 . the law code stele of hammurabi ( essay , video , images , additional resources ) 20 . temple of amun-re and hypostyle hall ( essay , video , images , additional resources ) 21 . mortuary temple of hatshepsut ( video , images , additional resources ) 22 . akhenaton , nefertiti , and three daughters ( video , images , additional resources ) 23 . tutankhamun 's tomb , innermost coffin ( essay , images , additional resources ) 24 . last judgment of hu-nefer , ( book of the dead ) ( video , essay , images , additional resources ) 25 . lamassu from the citadel of sargon ii , dur sharrukin ( modern iraq ) ( video , images , additional resources ) 26 . athenian agora ( video , images , additional resources ) 27 . anavysos kouros ( video , images , additional resources ) 28 . peplos kore from the acropolis ( video , images , additional resources ) 29 . sarcophagus of the spouses ( video , essay , images , additional resources ) 30 . audience hall ( apadana ) of darius and xeres ( essay , images , additional resources ) related : column capital , audience hall ( apadana ) of darius at susa ( video , images ) 31 . temple of minerva ( veii near rome , italy ) , sculpture of apollo ( essay , video , images , additional resources ) 32 . tomb of the triclinium ( essay , images , additional resources ) 33 . niobides krater ( video , images , additional resources ) 34 . doryphoros ( spear bearer ) ( video , essay , images , additional resources ) 35 . acropolis a. parthenon ( video , essay , images , additional resources ) i. helios , horses , dionysus ( heracles ? ) , east pediment ( video , images , additional resources ) ii . frieze ( video , images ) , plaque of the ergastines ( video , images , additional resources ) b . victory adjusting her sandel , temple of athena nike ( video , images , additional resources ) 36 . grave stele of hegeso ( video , images , additional resources ) 37 . winged victory of samothrace ( video , images , additional resources ) 38 . great altar of zeus and athena at pergamon ( video , images , additional resources 39 . house of the vettii ( essay , images , additional resources ) 40 . alexander mosaic from the house of the faun , pompeii ( video , images , additional resources ) 41 . seated boxer ( video , images , additional resources ) 42 . head of a roman patrician ( essay , images , additional resources ) related : veristic male portrait ( video , additional resources ) 43 . augustus of prima porta ( video , essay , images , additional resources ) 44 . colosseum ( flavian amphitheater ) ( video , images , additional resources 45 . forum of trajan ( essay , additional resources ) a . forum ( video , images , additional resources ) b . column ( video , images , additional resources ) c. markets ( video , images , additional resources ) 46 . pantheon ( video , images , additional resources ) 47 . ludovisi battle sarcophagus ( video , images , additional resources ) content area 3 early europe and colonial americas 200-1750 c.e . 48 . catacomb of priscilla ( greek chapel , orant , good shepherd frescos ) ( video , images , additional resources ) 49 . santa sabina ( video , essay , images , additional resources ) 50 . vienna genesis b. jacob wrestling the angel ( video , images , additional resources ) a. rebecca and eliezer at the well ( essay , images ) 51 . san vitale ( including justinian and theodora panels ) ( video , essay , images , additional resources ) 52 . hagia sophia ( video , essay , images , additional resources ) a. theotokos mosaic ( video , images , additional resources ) b. deësis mosaic ( video , images , additional resources ) c. hagia sophia as a mosque ( video , images , additional resources ) 53 . merovingian looped fibulae ( essay , images , additional resources additional resources 2 ) 54 . virgin ( theotokos ) and child between saints theodore and george ( essay , images , additional resources ) 55 . lindisfarne gospels , st. matthew , cross-carpet page ; st. luke incipit page ( essay , images , additional resources ) 56 . great mosque , córdoba , spain ( essay , images , additional resources ) 57 . pyxis of al-mughira ( essay , images , additional resources ) 58 . church of sainte-foy and reliquary ( essay , images , additional resources ) 59 . bayeux tapestry ( essay , images , additional resources ) 60 . chartres cathedral ( video , image , additional resources ) 61 . dedication page with blanche of castile and king louis ix of france , essay , image , additional resources ) and scenes from the apocalypse ( essay , image , additional resources ) —both from bibles moralisée ( moralized bibles ) 62 . röttgen pietà ( video , essay , images ) 63 . arena ( scrovegni ) chapel , including lamentation ( additional resources ) a . introduction ( video ) b. fresco cycle ( video ) c. lamentation ( video ) d. last judgment ( video ) 64 . golden haggadah ( essay , images , additional resources 1 , additional resources 2 ) 65 . alhambra ( essay , additional resources ) 66 . annunciation triptych ( merode altarpiece ) ( video , images , additional resources ) 67 . pazzi chapel , filipo brunellschi ( video , images , additional resources ) 68 . the arnolfini portrait , jan van eyck ( video , additional resources ) 69 . david , donatello ( video , essay , images , additional resources ) 70 . palazzo rucellai , leon battista alberti ( video , essay , additional resources ) 71 . madonna and child with two angels , fra filippo lippi ( video , image , additionalresources ) 72 . birth of venus , sandro botticelli ( video , image , additional resources ) 73 . last supper , leonardo da vinci ( video , essay , image , additional resources 75 . sistine chapel ceiling and altar wall frescos , michelangelo a. ceiling ( video , essay , study for sibyl video , additional resources ) b. altar wall ( video , additional resources ) 76 . school of athens , raphael ( video , essay , images , additional resources ) 77.isenheim altarpiece , matthias grünewald ( essay , images , additional resources ) 78 . entombment of christ , jacobo da pontormo ( video , image , , additional resources ) 79 . allegory of law and grace , lucas cranach the elder ( essay , images , additionalresources ) 80 . venus of urbino , titian ( video , image , additional resources ) 81 . frontispiece of the codex mendoza ( essay , images , additional resources 82 . il gesù , including triumph of the name of jesus ceiling fresco ( video , image 1 , image 2 , additional resources ) 83 . hunters in the snow , pieter bruegel the elder ( video , image , additional resources ) 84 . mosque of selim ii ( essay , images additional resources ) 85 . calling of saint matthew , caravaggio ( video , images , additional resources ) 86 . henri iv receives the portrait of marie de'medici , from the marie de'medici cycle , peter paul rubens ( video , essay , images , additional resources ) 87 . self-portrait with saskia , rembrandt van rijn ( essay , image , additional resources ) 88 . san carlo alle quattro fontane , francesco borromini ( video , images , additionalresources ) 89 . ecstasy of saint teresa , gian lorenzo bernini ( video , images , additional resources ) 90 . angel with arquebus , asiel timor dei , master of calamarca ( essay , image , additional resources ) 91 . las meninas , diego velazquez ( video , image , additional resources ) 92 . woman holding a balance , johannes vermeer ( video , image , additional resources ) 93 . the palace at versailles ( essay , images and maps , additional resources ) 94 . screen with siege of belgrade and hunting scene ( video , images , additional resources ) 95 . the virgin of guadalupe ( virgen de guadalupe ) _ , miguel gonzález ( video , essay , additional resources ) 96 . fruit and insects , rachel ruysch ( video , images , additional resources ) 97 . spaniard and indian produce a mestizo , attributed to juan rodríguez juárez ( essay , image , additional resources ) 98 . the tête à tête , from marriage a la mode , william hogarth ( video , images , additional resources ) content area 4 later europe and americas 1750-1980 c.e . 99 . portrait of sor juana inés de la cruz , miguel cabrera ( essay , [ additional resources ] [ http : //jps.library.utoronto.ca/index.php/emw/article/viewfile/18743/15678 ] ) 100 . a philosopher giving a lecture on the orrery , joseph wright of derby ( essay , image , additional resources ) 101 . the swing , jean-honoré fragonard ( video , images , additional resources ) 102 . monticello , thomas jefferson ( essay , images additional resources 1 , 2 ) 103 . the oath of the horatii , jacques-louis david ( video , essay , images , additionalresources ) 104 . george washington , jean-antoine houdon ( essay , image , additional resources 1 , 2 ) 105 . self-portrait , elisabeth louise vigée-lebrun ( essay , image , additional resources ) 106 . y no hai remedio ( and there 's nothing to be done ) , from los desastres de la guerra ( the disasters of war ) , plate 15 , francesco de goya ( essay , images , additional resources 107 . la grande odalisque , jean-auguste-dominique ingres ( video , essay , images , additional resources ) 108 . liberty leading the people , eugène delacroix ( video , essay , images , additionalresources ) 109 . view from mount holyoke , northampton , massachusetts , after athunderstorm—the oxbow , thomas cole ( video , essay , images , additional resources ) 110 . still life in studio , louis-jacques-mandé daguerre ( essay , image , additional resources ) 111 . slave ship ( slavers throwing overboard the dead and dying , typhoon coming on ) , joseph mallord william turner ( video , images , additional resources ) 112 . palace of westminster ( houses of parliament ) , charles barry , a.w.n . pugin ( video , images , additional resources ) 113 . the stonebreakers , gustave courbet ( essay , image , additional resources 1 , 2 ) 114 . nadar elevating photography to art , honoré daumier ( essay , image , additionalresources ) 115 . olympia , édouard manet ( video , image , additional resources ) 116 . the saint-lazare station , claude monet ( video , image , additional resources 117 . the horse in motion , eadweard muybridge ( essay , image , additional resources ) 118 . the valley of mexico from the hillside of santa isabel , josé maría velasco ( video , essay , photos ) 119 . the burghers of calais , auguste rodin ( essay , image , 3d image , additional resources ) 120 . the starry night , vincent van gogh ( essay , image , additional resources ) 121 . the coiffure , mary cassatt ( essay , image , additional resources ) 122 . the scream , edvard munch ( essay , image , additional resources ) 123 . where do we come from ? what are we ? where are we going ? , paul gauguin ( essay , images , additional resources ) 124 . carson , pirie , scott and company building , louis sullivan ( essay , images , additionalresources ) 125 . mont sainte-victoire , paul cézanne ( video , essay , image , additional resources 126 . les demoiselles d'avignon , pablo picasso ( video , images , additional resources 127 . the steerage , alfred stieglitz ( essay , image , additional resources ) 128 . the kiss , gustav klimt ( video , image , additional resources ) 129 . the kiss , constantin brancusi ( video , images , additional resources ) 130 . the portuguese , georges braque ( essay , images , additional resources ) 131 . the goldfish , henri matisse ( essay , image , additional resources ) 132 . improvisation 28 ( second version ) , vasily kandinsky ( video , photo , additional resources 133 . self-portrait as a soldier , ernst ludwig kirchner ( essay , image , additional resources 134 . memorial sheet of karl liebknecht , käthe kollwitz ( essay , image , additional resources ) 135 . villa savoye , le corbusier ( essay , images , additional resources ) 136 . composition with red , blue and yellow , piet mondrian ( essay , image , additional resources ) 137 . illustration from the results of the first five-year plan , varvarastepanova ( essay , image , additional resources ) 138 . object ( le déjeuner en fourrure ) , meret oppenheim ( essay , quiz , image , additional resources 139 . fallingwater , frank lloyd wright ( essay , plans and elevations , images , additionalresources ) 140 . the two fridas , frida kahlo ( essay , image , additional resources ) 141 . the migration of the negro , panel no . 49 , jacob lawrence ( video-short version , video-long version , photo , additional resources ) 142 . the jungle , wilfredo lam ( essay , image , additional resources ) 143 . dream of a sunday afternoon in alameda central park , diego rivera ( essay , image , additional resources ) 144 . fountain , marcel duchamp ( video , images , additional resources ) 145 . woman i , willem de kooning ( video , images , additional resources ) 146 . seagram building , ludwig mies van der rohe , philip johnson ( video , images , additional resources ) 147 . marilyn diptych , andy warhol ( essay , image , additional resources ) 148 . narcissus garden , yayoi kusama ( essay , image , additional resources ) 149 . the bay , helen frankenthaler ( essay , quiz , image , additional resources ) 150 . lipstick ( ascending ) on caterpillar tracks , claes oldenburg ( essay , photo , additional resoures ) 151 . spiral jetty , robert smithson ( video , images , additional resources ) 152 . house in new castle county , robert ventura , john rausch and denise scott brown ( essay , photos , additional resources ) content area 5 indigenous americas 1000 b.c.e.-1980 c.e . 153 . chavín de huántar ( essay , image , additional resources ) 154 . mesa verde cliff dwellings ( essay , video , photos , additional resources ) 155 . yaxchilán lintel 25 , structure 23 ( essay , essay on related lintel , photo , additional resources ) 156 . great serpent mound ( essay , photos , additional resources ) 157 . templo mayor , main aztec temple ( video , essay , images , additional resources ) a . the coyolxauhqui stone ( video , images , additional resources ) b . calendar stone ( video , images , additional resources ) c. olmec-style mask ( video , images , additional resources ) 158 . ruler 's feather headdress ( probably of moctezuma ii ) ( video , images , additionalresources , see page 12 ) 159 . city of cusco ( essay , video , photos , additional resources ) 160 . maize cobs ( essay , image , additional resources 1 , 2 ) 161 . city of machu picchu ( essay , video , photos , additional resources ) 162 . all-t'oqapu tunic ( essay , photo , additional resources ) 163 . bandolier bag ( essay , photo , additional resources ) 164 . transformation mask ( essay , image , additional resources ) 165 . painted elk hide , attributed to cotsiogo ( cadzi cody ) ( essay , image , additionalresources ) 166 . black-on-black ceramic vessel , maria martínez and julian martínez ( essay , image , additional resources ) content area 6 africa 1100-1980 c.e . 167 . conical tower and circular wall of great zimbabwe ( video , images , additional resources ) 168 . great mosque of djenné ( essay , video , images , additional resources ) 169 . wall plaque , from oba 's palace ( essay , images , additional resources ) 170 . sika dwa kofi ( golden stool ) ( video , image , additional resources ) 171 . ndop ( portrait figure ) of king mishe mishyaang mambul ( essay , image , additional resources ) 172 . nkisi n ’ kondi ( essay , image , additional resources ) 173 . female ( pwo ) mask ( video , images , additional resources ) 174 . portrait mask ( mblo ) ( essay , image , additional resources ) 175 . bundu mask ( video , image , additional resources ) 176 . ikenga ( shrine figure ) ( video , image , additional resources ) 177 . lukasa ( memory board ) ( essay , image , additional resources ) 178 . aka elephant mask ( video , photos , additional resources 179 . reliquary figure ( byeri ) ( video , photos , additional resources ) 180 . veranda post of enthroned king and senior wife ( opo ogoga ) ( related video , image 1 , image 2 , additional resources ) content area 7 west and central asia 500 b.c.e.-1980 c.e . 181 . petra , jordan : treasury and great temple ( additional resources ) a. nabataeans introduction ( essay ) b. petra and the treasury ( essay , images ) c. petra and the great temple ( essay , images ) d. unesco siq project ( video ) 182 . buddha , bamiyan ( essay , image , video , additional resources 1 , 2 ) 183 . the kaaba ( essay , images , additional resources ) 184 . jowo rinpoche , enshrined in the jokhang temple ( essay , images , additional resources ) 185 . dome of the rock ( essay , images , additional resources ) 186 . great mosque ( masjid-e jameh ) , isfahan ( essay , images , additional resources ) 187 . folio from a qur'an ( essay , image , additional resources ) 188 . basin ( baptistère de saint louis ) , mohammed ibn al-zain ( video , images , additional resources ) 189 . bahram gur fights the karg , folio from the great il-khanid shahnama ( essay , image , additional resources ) 190 . the court of gayumars , folio from shah tahmasp's shahnama ( essay , image , additional resources ) 191 . the ardabil carpet ( essay , images , additional resources ) content area 8 south , east and southeast asia 300 b.c.e.-1980 c.e . 192 . great stupa at sanchi ( essay , video , images , additional resources ) 193 . terracotta warriors from mausoleum of the first qin emperor of china ( essay , video , images , additional resources ) 194 . funeral banner of lady dai ( xin zhui ) ( essay , images , additional resources 195 . longmen caves ( essay , video , images , additional resources ) 196 . gold and jade crown ( essay , image , additional resources ) 197 . todai-ji ( essay , video , images , additional resources ) 198 . borobudur ( essay , images , additional resources ) 199 . angkor , the temple of angkor wat , the city of angkor thom , cambodia ( essay , video , images , additional resources ) 200 . lakshmana temple ( essay , additional resources ) 201 . travelers among mountains and streams , fan kuan ( essay , image , additional resources ) 202 . shiva as lord of dance ( nataraja ) ( essay , image , additional resources ) 203 . night attack on the sanjô palace ( essay , photos , additional resources ) 204 . the david vases ( video , essay , images , additional resources 1 , 2 ) 205 . portrait of sin sukju ( essay , image , additional resources ) 206 . forbidden city ( essay , video , images , additional resources ) 207 . ryoan-ji ( video , essay , images , additional resources 1 , 2 ) 208 . jahangir preferring a sufi shaikh to kings , bichitr ( essay , image , additional resources ) 209 . taj mahal ( essay , images , additional resources ) 210 . white and red plum blossoms , ogata korin ( essay , image , additional resources 1 , additional resources 2 , additional resources 3 ) 211 . under the wave off kanagawa ( kanagawa oki nami ura ) , also known as the great wave , from the series thirty-six views of mount fuji , katsushika hokusai ( essay , image , additional resources ) 212 . chairman mao en route to anyuan ( essay , image additional resources ) content area 9 the pacific 700-1980 c.e . 213 . nan madol 214 . moai on platform ( ahu ) ( video , essay , images , additional resources 1 , 2 , 3 ) 215 . 'ahu 'ula ( feather cape ) ( essay , image , additional resources ) 216 . staff god ( essay , images , additional resources ) 217 . female deity from nukuoro ( essay , images , additional resources 1 , 2 , 3 ) 218 . buk mask ( video , images , additional resources ) 219 . hiapo ( tapa ) ( essay , image , additional resources ) 220 . tamati waka nene , gottfried lindaur ( essay , image , additional resources 221 . navigation chart ( essay , image , additional resources ) 222 . malagan display and mask ( essay , image 1 , 2 , additional resources ) 223 . presentation of fijian mats and tapas cloths to queen elizabeth ii ( essay , image , additional resources 1 , 2 ) content area 10 global contemporary 1980 c.e . to present 224 . the gates , christo and jeanne-claude ( essay , images , additional resources ) 225 . vietnam veterans memorial , maya lin ( video , images , additional resources ) 226 . horn players , jean-michel basquiat ( essay , image , additional resources ) 227 . summer trees , song su-nam ( essay , image , additional resources ) 228 . androgyne iii , magdalena abakanowicz ( essay , images , additional resources ) 229 . a book from the sky , xu bing ( video , images , additional resources 230 . pink panther , jeff koons ( essay , images , additional resources ) 231 . untitled ( # 228 ) , from the history portraits series , cindy sherman ( essay , image , additional resources ) 232 . dancing at the louvre , from the series , the french collection , part 1 ; # 1 , faith ringgold ( essay , image , additional resources ) 233 . trade ( gifts for trading land with white people ) , jaune quick-to-see smith ( essay , image , additional resources ) 234 . earth ’ s creation , emily kame kngwarreye ( essay , image , additional resources ) 235 . rebellious silence , from the women of allah series , shirin neshat ( artist ) ; photo by cynthia preston ( essay , image , additional resources ) 236 . en la barberia no se llora ( no crying allowed in the barbershop ) , pepon osorio ( essay , image 1 , 2 , additional resources ) 237 . pisupo lua afe ( corned beef 2000 ) , michel tuffery ( essay , artist , biography , video 1 , video 2 , images ) 238 . electronic superhighway , nam june paik ( essay , image , additional resources 239 . the crossing , bill viola ( essay , image , additional resources ) 240 . guggenheim museum bilbao , frank gehry ( essay , images , additional resources ) 241 . pure land , mariko mori ( essay , image , additional resources ) 242 . lying with the wolf , kiki smith ( essay , image , additional resources ) 243.darkytown rebellion , kara walker ( essay , image , additional resources ) 244 . the swing ( after fragonard ) , yinka shonibare ( essay , images , additional resources ) 245 . old man ’ s cloth , el anatsui ( video , essay , image , additional resources ) 246 . stadia ii , julie mehretu ( essay , image , additional resources ) 247 . preying mantra , wangechi mutu ( essay , image , additional resources ) 248 . shibboleth , doris salcedo ( essay , images , additional resources ) 249 . maxxi national museum of xxi century arts , zaha hadid ( video , images , additional resources ) 250 . kui hua zi ( sunflower seeds ) , ai weiwei ( essay , video , images , additional resources ) special thanks to the many art historians and curators who have contributed their expertise as well as our museum partners , the american museum of natural history , the asian art museum , the british museum , the j. paul getty museum , the metropolitan museum of art , the museum of modern art , and tate . ap art history is a registered trademark of the college board , which was not involved in the production of , and does not endorse , this product .
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for more context on these works of art , please see the art history sections of khan academy . content area 1 global prehistory 30,000-500b.c.e . 1 .
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are the content areas supposed to be studied and completed every week on their own , or should they take longer than that ?
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this list links to videos , essays , images , and additional resources for the 250 required works of art for the ap* art history course and exam . smarthistory is adding new materials regularly . for more context on these works of art , please see the art history sections of khan academy . content area 1 global prehistory 30,000-500b.c.e . 1 . apollo 11 stones ( essay , image , additional resources ) 2 . great hall of bulls ( essay , image , additional resources ) 3 . camelid sacrum in the shape of a canine ( essay , image , additional resources ) 4 . running horned woman ( essay , image , additional resources ) 5 . bushel with ibex motifs ( video , images , additional resources ) 6 . anthropomorphic stele ( essay , image , additional resources ) 7 . jade cong ( video , article , image , additional resources ) 8 . stonehenge ( essay , video , image 1 , image 2 , additional resources ) 9 . ambum stone ( essay , image , additional resources ) 10 . tlatilco female figurine ( video , essay , image , additional resources ) 11 . terracotta fragment ( lapita ) ( essay , image , additional resources ) content area 2 ancient mediterranean 3,500-300 b.c.e . 12 . white temple and its ziggurat ( essay , image 1 , image 2 additional resources ) 13 . palette of king narmer ( essay , images , additional resources ) 14 . statues of votive figures , from the square temple at eshnunna ( video , images , additional resources ) 15 . seated scribe ( video , images , additional resources ) 16 . standard of ur from the royal tombs at ur ( video , essay , images , additional resources 17 . great pyramids of giza ( essay , images , additional resources ) a. pyramid of khufu ( essay , additional resources ) b. pyramid of khafre and the great sphinx ( essay , additional resources ) c. pyramid of menkaura ( essay , additional resources ) 18 . king menkaura and queen ( essay , images , additional resources ) 19 . the law code stele of hammurabi ( essay , video , images , additional resources ) 20 . temple of amun-re and hypostyle hall ( essay , video , images , additional resources ) 21 . mortuary temple of hatshepsut ( video , images , additional resources ) 22 . akhenaton , nefertiti , and three daughters ( video , images , additional resources ) 23 . tutankhamun 's tomb , innermost coffin ( essay , images , additional resources ) 24 . last judgment of hu-nefer , ( book of the dead ) ( video , essay , images , additional resources ) 25 . lamassu from the citadel of sargon ii , dur sharrukin ( modern iraq ) ( video , images , additional resources ) 26 . athenian agora ( video , images , additional resources ) 27 . anavysos kouros ( video , images , additional resources ) 28 . peplos kore from the acropolis ( video , images , additional resources ) 29 . sarcophagus of the spouses ( video , essay , images , additional resources ) 30 . audience hall ( apadana ) of darius and xeres ( essay , images , additional resources ) related : column capital , audience hall ( apadana ) of darius at susa ( video , images ) 31 . temple of minerva ( veii near rome , italy ) , sculpture of apollo ( essay , video , images , additional resources ) 32 . tomb of the triclinium ( essay , images , additional resources ) 33 . niobides krater ( video , images , additional resources ) 34 . doryphoros ( spear bearer ) ( video , essay , images , additional resources ) 35 . acropolis a. parthenon ( video , essay , images , additional resources ) i. helios , horses , dionysus ( heracles ? ) , east pediment ( video , images , additional resources ) ii . frieze ( video , images ) , plaque of the ergastines ( video , images , additional resources ) b . victory adjusting her sandel , temple of athena nike ( video , images , additional resources ) 36 . grave stele of hegeso ( video , images , additional resources ) 37 . winged victory of samothrace ( video , images , additional resources ) 38 . great altar of zeus and athena at pergamon ( video , images , additional resources 39 . house of the vettii ( essay , images , additional resources ) 40 . alexander mosaic from the house of the faun , pompeii ( video , images , additional resources ) 41 . seated boxer ( video , images , additional resources ) 42 . head of a roman patrician ( essay , images , additional resources ) related : veristic male portrait ( video , additional resources ) 43 . augustus of prima porta ( video , essay , images , additional resources ) 44 . colosseum ( flavian amphitheater ) ( video , images , additional resources 45 . forum of trajan ( essay , additional resources ) a . forum ( video , images , additional resources ) b . column ( video , images , additional resources ) c. markets ( video , images , additional resources ) 46 . pantheon ( video , images , additional resources ) 47 . ludovisi battle sarcophagus ( video , images , additional resources ) content area 3 early europe and colonial americas 200-1750 c.e . 48 . catacomb of priscilla ( greek chapel , orant , good shepherd frescos ) ( video , images , additional resources ) 49 . santa sabina ( video , essay , images , additional resources ) 50 . vienna genesis b. jacob wrestling the angel ( video , images , additional resources ) a. rebecca and eliezer at the well ( essay , images ) 51 . san vitale ( including justinian and theodora panels ) ( video , essay , images , additional resources ) 52 . hagia sophia ( video , essay , images , additional resources ) a. theotokos mosaic ( video , images , additional resources ) b. deësis mosaic ( video , images , additional resources ) c. hagia sophia as a mosque ( video , images , additional resources ) 53 . merovingian looped fibulae ( essay , images , additional resources additional resources 2 ) 54 . virgin ( theotokos ) and child between saints theodore and george ( essay , images , additional resources ) 55 . lindisfarne gospels , st. matthew , cross-carpet page ; st. luke incipit page ( essay , images , additional resources ) 56 . great mosque , córdoba , spain ( essay , images , additional resources ) 57 . pyxis of al-mughira ( essay , images , additional resources ) 58 . church of sainte-foy and reliquary ( essay , images , additional resources ) 59 . bayeux tapestry ( essay , images , additional resources ) 60 . chartres cathedral ( video , image , additional resources ) 61 . dedication page with blanche of castile and king louis ix of france , essay , image , additional resources ) and scenes from the apocalypse ( essay , image , additional resources ) —both from bibles moralisée ( moralized bibles ) 62 . röttgen pietà ( video , essay , images ) 63 . arena ( scrovegni ) chapel , including lamentation ( additional resources ) a . introduction ( video ) b. fresco cycle ( video ) c. lamentation ( video ) d. last judgment ( video ) 64 . golden haggadah ( essay , images , additional resources 1 , additional resources 2 ) 65 . alhambra ( essay , additional resources ) 66 . annunciation triptych ( merode altarpiece ) ( video , images , additional resources ) 67 . pazzi chapel , filipo brunellschi ( video , images , additional resources ) 68 . the arnolfini portrait , jan van eyck ( video , additional resources ) 69 . david , donatello ( video , essay , images , additional resources ) 70 . palazzo rucellai , leon battista alberti ( video , essay , additional resources ) 71 . madonna and child with two angels , fra filippo lippi ( video , image , additionalresources ) 72 . birth of venus , sandro botticelli ( video , image , additional resources ) 73 . last supper , leonardo da vinci ( video , essay , image , additional resources 75 . sistine chapel ceiling and altar wall frescos , michelangelo a. ceiling ( video , essay , study for sibyl video , additional resources ) b. altar wall ( video , additional resources ) 76 . school of athens , raphael ( video , essay , images , additional resources ) 77.isenheim altarpiece , matthias grünewald ( essay , images , additional resources ) 78 . entombment of christ , jacobo da pontormo ( video , image , , additional resources ) 79 . allegory of law and grace , lucas cranach the elder ( essay , images , additionalresources ) 80 . venus of urbino , titian ( video , image , additional resources ) 81 . frontispiece of the codex mendoza ( essay , images , additional resources 82 . il gesù , including triumph of the name of jesus ceiling fresco ( video , image 1 , image 2 , additional resources ) 83 . hunters in the snow , pieter bruegel the elder ( video , image , additional resources ) 84 . mosque of selim ii ( essay , images additional resources ) 85 . calling of saint matthew , caravaggio ( video , images , additional resources ) 86 . henri iv receives the portrait of marie de'medici , from the marie de'medici cycle , peter paul rubens ( video , essay , images , additional resources ) 87 . self-portrait with saskia , rembrandt van rijn ( essay , image , additional resources ) 88 . san carlo alle quattro fontane , francesco borromini ( video , images , additionalresources ) 89 . ecstasy of saint teresa , gian lorenzo bernini ( video , images , additional resources ) 90 . angel with arquebus , asiel timor dei , master of calamarca ( essay , image , additional resources ) 91 . las meninas , diego velazquez ( video , image , additional resources ) 92 . woman holding a balance , johannes vermeer ( video , image , additional resources ) 93 . the palace at versailles ( essay , images and maps , additional resources ) 94 . screen with siege of belgrade and hunting scene ( video , images , additional resources ) 95 . the virgin of guadalupe ( virgen de guadalupe ) _ , miguel gonzález ( video , essay , additional resources ) 96 . fruit and insects , rachel ruysch ( video , images , additional resources ) 97 . spaniard and indian produce a mestizo , attributed to juan rodríguez juárez ( essay , image , additional resources ) 98 . the tête à tête , from marriage a la mode , william hogarth ( video , images , additional resources ) content area 4 later europe and americas 1750-1980 c.e . 99 . portrait of sor juana inés de la cruz , miguel cabrera ( essay , [ additional resources ] [ http : //jps.library.utoronto.ca/index.php/emw/article/viewfile/18743/15678 ] ) 100 . a philosopher giving a lecture on the orrery , joseph wright of derby ( essay , image , additional resources ) 101 . the swing , jean-honoré fragonard ( video , images , additional resources ) 102 . monticello , thomas jefferson ( essay , images additional resources 1 , 2 ) 103 . the oath of the horatii , jacques-louis david ( video , essay , images , additionalresources ) 104 . george washington , jean-antoine houdon ( essay , image , additional resources 1 , 2 ) 105 . self-portrait , elisabeth louise vigée-lebrun ( essay , image , additional resources ) 106 . y no hai remedio ( and there 's nothing to be done ) , from los desastres de la guerra ( the disasters of war ) , plate 15 , francesco de goya ( essay , images , additional resources 107 . la grande odalisque , jean-auguste-dominique ingres ( video , essay , images , additional resources ) 108 . liberty leading the people , eugène delacroix ( video , essay , images , additionalresources ) 109 . view from mount holyoke , northampton , massachusetts , after athunderstorm—the oxbow , thomas cole ( video , essay , images , additional resources ) 110 . still life in studio , louis-jacques-mandé daguerre ( essay , image , additional resources ) 111 . slave ship ( slavers throwing overboard the dead and dying , typhoon coming on ) , joseph mallord william turner ( video , images , additional resources ) 112 . palace of westminster ( houses of parliament ) , charles barry , a.w.n . pugin ( video , images , additional resources ) 113 . the stonebreakers , gustave courbet ( essay , image , additional resources 1 , 2 ) 114 . nadar elevating photography to art , honoré daumier ( essay , image , additionalresources ) 115 . olympia , édouard manet ( video , image , additional resources ) 116 . the saint-lazare station , claude monet ( video , image , additional resources 117 . the horse in motion , eadweard muybridge ( essay , image , additional resources ) 118 . the valley of mexico from the hillside of santa isabel , josé maría velasco ( video , essay , photos ) 119 . the burghers of calais , auguste rodin ( essay , image , 3d image , additional resources ) 120 . the starry night , vincent van gogh ( essay , image , additional resources ) 121 . the coiffure , mary cassatt ( essay , image , additional resources ) 122 . the scream , edvard munch ( essay , image , additional resources ) 123 . where do we come from ? what are we ? where are we going ? , paul gauguin ( essay , images , additional resources ) 124 . carson , pirie , scott and company building , louis sullivan ( essay , images , additionalresources ) 125 . mont sainte-victoire , paul cézanne ( video , essay , image , additional resources 126 . les demoiselles d'avignon , pablo picasso ( video , images , additional resources 127 . the steerage , alfred stieglitz ( essay , image , additional resources ) 128 . the kiss , gustav klimt ( video , image , additional resources ) 129 . the kiss , constantin brancusi ( video , images , additional resources ) 130 . the portuguese , georges braque ( essay , images , additional resources ) 131 . the goldfish , henri matisse ( essay , image , additional resources ) 132 . improvisation 28 ( second version ) , vasily kandinsky ( video , photo , additional resources 133 . self-portrait as a soldier , ernst ludwig kirchner ( essay , image , additional resources 134 . memorial sheet of karl liebknecht , käthe kollwitz ( essay , image , additional resources ) 135 . villa savoye , le corbusier ( essay , images , additional resources ) 136 . composition with red , blue and yellow , piet mondrian ( essay , image , additional resources ) 137 . illustration from the results of the first five-year plan , varvarastepanova ( essay , image , additional resources ) 138 . object ( le déjeuner en fourrure ) , meret oppenheim ( essay , quiz , image , additional resources 139 . fallingwater , frank lloyd wright ( essay , plans and elevations , images , additionalresources ) 140 . the two fridas , frida kahlo ( essay , image , additional resources ) 141 . the migration of the negro , panel no . 49 , jacob lawrence ( video-short version , video-long version , photo , additional resources ) 142 . the jungle , wilfredo lam ( essay , image , additional resources ) 143 . dream of a sunday afternoon in alameda central park , diego rivera ( essay , image , additional resources ) 144 . fountain , marcel duchamp ( video , images , additional resources ) 145 . woman i , willem de kooning ( video , images , additional resources ) 146 . seagram building , ludwig mies van der rohe , philip johnson ( video , images , additional resources ) 147 . marilyn diptych , andy warhol ( essay , image , additional resources ) 148 . narcissus garden , yayoi kusama ( essay , image , additional resources ) 149 . the bay , helen frankenthaler ( essay , quiz , image , additional resources ) 150 . lipstick ( ascending ) on caterpillar tracks , claes oldenburg ( essay , photo , additional resoures ) 151 . spiral jetty , robert smithson ( video , images , additional resources ) 152 . house in new castle county , robert ventura , john rausch and denise scott brown ( essay , photos , additional resources ) content area 5 indigenous americas 1000 b.c.e.-1980 c.e . 153 . chavín de huántar ( essay , image , additional resources ) 154 . mesa verde cliff dwellings ( essay , video , photos , additional resources ) 155 . yaxchilán lintel 25 , structure 23 ( essay , essay on related lintel , photo , additional resources ) 156 . great serpent mound ( essay , photos , additional resources ) 157 . templo mayor , main aztec temple ( video , essay , images , additional resources ) a . the coyolxauhqui stone ( video , images , additional resources ) b . calendar stone ( video , images , additional resources ) c. olmec-style mask ( video , images , additional resources ) 158 . ruler 's feather headdress ( probably of moctezuma ii ) ( video , images , additionalresources , see page 12 ) 159 . city of cusco ( essay , video , photos , additional resources ) 160 . maize cobs ( essay , image , additional resources 1 , 2 ) 161 . city of machu picchu ( essay , video , photos , additional resources ) 162 . all-t'oqapu tunic ( essay , photo , additional resources ) 163 . bandolier bag ( essay , photo , additional resources ) 164 . transformation mask ( essay , image , additional resources ) 165 . painted elk hide , attributed to cotsiogo ( cadzi cody ) ( essay , image , additionalresources ) 166 . black-on-black ceramic vessel , maria martínez and julian martínez ( essay , image , additional resources ) content area 6 africa 1100-1980 c.e . 167 . conical tower and circular wall of great zimbabwe ( video , images , additional resources ) 168 . great mosque of djenné ( essay , video , images , additional resources ) 169 . wall plaque , from oba 's palace ( essay , images , additional resources ) 170 . sika dwa kofi ( golden stool ) ( video , image , additional resources ) 171 . ndop ( portrait figure ) of king mishe mishyaang mambul ( essay , image , additional resources ) 172 . nkisi n ’ kondi ( essay , image , additional resources ) 173 . female ( pwo ) mask ( video , images , additional resources ) 174 . portrait mask ( mblo ) ( essay , image , additional resources ) 175 . bundu mask ( video , image , additional resources ) 176 . ikenga ( shrine figure ) ( video , image , additional resources ) 177 . lukasa ( memory board ) ( essay , image , additional resources ) 178 . aka elephant mask ( video , photos , additional resources 179 . reliquary figure ( byeri ) ( video , photos , additional resources ) 180 . veranda post of enthroned king and senior wife ( opo ogoga ) ( related video , image 1 , image 2 , additional resources ) content area 7 west and central asia 500 b.c.e.-1980 c.e . 181 . petra , jordan : treasury and great temple ( additional resources ) a. nabataeans introduction ( essay ) b. petra and the treasury ( essay , images ) c. petra and the great temple ( essay , images ) d. unesco siq project ( video ) 182 . buddha , bamiyan ( essay , image , video , additional resources 1 , 2 ) 183 . the kaaba ( essay , images , additional resources ) 184 . jowo rinpoche , enshrined in the jokhang temple ( essay , images , additional resources ) 185 . dome of the rock ( essay , images , additional resources ) 186 . great mosque ( masjid-e jameh ) , isfahan ( essay , images , additional resources ) 187 . folio from a qur'an ( essay , image , additional resources ) 188 . basin ( baptistère de saint louis ) , mohammed ibn al-zain ( video , images , additional resources ) 189 . bahram gur fights the karg , folio from the great il-khanid shahnama ( essay , image , additional resources ) 190 . the court of gayumars , folio from shah tahmasp's shahnama ( essay , image , additional resources ) 191 . the ardabil carpet ( essay , images , additional resources ) content area 8 south , east and southeast asia 300 b.c.e.-1980 c.e . 192 . great stupa at sanchi ( essay , video , images , additional resources ) 193 . terracotta warriors from mausoleum of the first qin emperor of china ( essay , video , images , additional resources ) 194 . funeral banner of lady dai ( xin zhui ) ( essay , images , additional resources 195 . longmen caves ( essay , video , images , additional resources ) 196 . gold and jade crown ( essay , image , additional resources ) 197 . todai-ji ( essay , video , images , additional resources ) 198 . borobudur ( essay , images , additional resources ) 199 . angkor , the temple of angkor wat , the city of angkor thom , cambodia ( essay , video , images , additional resources ) 200 . lakshmana temple ( essay , additional resources ) 201 . travelers among mountains and streams , fan kuan ( essay , image , additional resources ) 202 . shiva as lord of dance ( nataraja ) ( essay , image , additional resources ) 203 . night attack on the sanjô palace ( essay , photos , additional resources ) 204 . the david vases ( video , essay , images , additional resources 1 , 2 ) 205 . portrait of sin sukju ( essay , image , additional resources ) 206 . forbidden city ( essay , video , images , additional resources ) 207 . ryoan-ji ( video , essay , images , additional resources 1 , 2 ) 208 . jahangir preferring a sufi shaikh to kings , bichitr ( essay , image , additional resources ) 209 . taj mahal ( essay , images , additional resources ) 210 . white and red plum blossoms , ogata korin ( essay , image , additional resources 1 , additional resources 2 , additional resources 3 ) 211 . under the wave off kanagawa ( kanagawa oki nami ura ) , also known as the great wave , from the series thirty-six views of mount fuji , katsushika hokusai ( essay , image , additional resources ) 212 . chairman mao en route to anyuan ( essay , image additional resources ) content area 9 the pacific 700-1980 c.e . 213 . nan madol 214 . moai on platform ( ahu ) ( video , essay , images , additional resources 1 , 2 , 3 ) 215 . 'ahu 'ula ( feather cape ) ( essay , image , additional resources ) 216 . staff god ( essay , images , additional resources ) 217 . female deity from nukuoro ( essay , images , additional resources 1 , 2 , 3 ) 218 . buk mask ( video , images , additional resources ) 219 . hiapo ( tapa ) ( essay , image , additional resources ) 220 . tamati waka nene , gottfried lindaur ( essay , image , additional resources 221 . navigation chart ( essay , image , additional resources ) 222 . malagan display and mask ( essay , image 1 , 2 , additional resources ) 223 . presentation of fijian mats and tapas cloths to queen elizabeth ii ( essay , image , additional resources 1 , 2 ) content area 10 global contemporary 1980 c.e . to present 224 . the gates , christo and jeanne-claude ( essay , images , additional resources ) 225 . vietnam veterans memorial , maya lin ( video , images , additional resources ) 226 . horn players , jean-michel basquiat ( essay , image , additional resources ) 227 . summer trees , song su-nam ( essay , image , additional resources ) 228 . androgyne iii , magdalena abakanowicz ( essay , images , additional resources ) 229 . a book from the sky , xu bing ( video , images , additional resources 230 . pink panther , jeff koons ( essay , images , additional resources ) 231 . untitled ( # 228 ) , from the history portraits series , cindy sherman ( essay , image , additional resources ) 232 . dancing at the louvre , from the series , the french collection , part 1 ; # 1 , faith ringgold ( essay , image , additional resources ) 233 . trade ( gifts for trading land with white people ) , jaune quick-to-see smith ( essay , image , additional resources ) 234 . earth ’ s creation , emily kame kngwarreye ( essay , image , additional resources ) 235 . rebellious silence , from the women of allah series , shirin neshat ( artist ) ; photo by cynthia preston ( essay , image , additional resources ) 236 . en la barberia no se llora ( no crying allowed in the barbershop ) , pepon osorio ( essay , image 1 , 2 , additional resources ) 237 . pisupo lua afe ( corned beef 2000 ) , michel tuffery ( essay , artist , biography , video 1 , video 2 , images ) 238 . electronic superhighway , nam june paik ( essay , image , additional resources 239 . the crossing , bill viola ( essay , image , additional resources ) 240 . guggenheim museum bilbao , frank gehry ( essay , images , additional resources ) 241 . pure land , mariko mori ( essay , image , additional resources ) 242 . lying with the wolf , kiki smith ( essay , image , additional resources ) 243.darkytown rebellion , kara walker ( essay , image , additional resources ) 244 . the swing ( after fragonard ) , yinka shonibare ( essay , images , additional resources ) 245 . old man ’ s cloth , el anatsui ( video , essay , image , additional resources ) 246 . stadia ii , julie mehretu ( essay , image , additional resources ) 247 . preying mantra , wangechi mutu ( essay , image , additional resources ) 248 . shibboleth , doris salcedo ( essay , images , additional resources ) 249 . maxxi national museum of xxi century arts , zaha hadid ( video , images , additional resources ) 250 . kui hua zi ( sunflower seeds ) , ai weiwei ( essay , video , images , additional resources ) special thanks to the many art historians and curators who have contributed their expertise as well as our museum partners , the american museum of natural history , the asian art museum , the british museum , the j. paul getty museum , the metropolitan museum of art , the museum of modern art , and tate . ap art history is a registered trademark of the college board , which was not involved in the production of , and does not endorse , this product .
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acropolis a. parthenon ( video , essay , images , additional resources ) i. helios , horses , dionysus ( heracles ? ) , east pediment ( video , images , additional resources ) ii . frieze ( video , images ) , plaque of the ergastines ( video , images , additional resources ) b . victory adjusting her sandel , temple of athena nike ( video , images , additional resources ) 36 .
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why are pictures/images in flickr ?
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this list links to videos , essays , images , and additional resources for the 250 required works of art for the ap* art history course and exam . smarthistory is adding new materials regularly . for more context on these works of art , please see the art history sections of khan academy . content area 1 global prehistory 30,000-500b.c.e . 1 . apollo 11 stones ( essay , image , additional resources ) 2 . great hall of bulls ( essay , image , additional resources ) 3 . camelid sacrum in the shape of a canine ( essay , image , additional resources ) 4 . running horned woman ( essay , image , additional resources ) 5 . bushel with ibex motifs ( video , images , additional resources ) 6 . anthropomorphic stele ( essay , image , additional resources ) 7 . jade cong ( video , article , image , additional resources ) 8 . stonehenge ( essay , video , image 1 , image 2 , additional resources ) 9 . ambum stone ( essay , image , additional resources ) 10 . tlatilco female figurine ( video , essay , image , additional resources ) 11 . terracotta fragment ( lapita ) ( essay , image , additional resources ) content area 2 ancient mediterranean 3,500-300 b.c.e . 12 . white temple and its ziggurat ( essay , image 1 , image 2 additional resources ) 13 . palette of king narmer ( essay , images , additional resources ) 14 . statues of votive figures , from the square temple at eshnunna ( video , images , additional resources ) 15 . seated scribe ( video , images , additional resources ) 16 . standard of ur from the royal tombs at ur ( video , essay , images , additional resources 17 . great pyramids of giza ( essay , images , additional resources ) a. pyramid of khufu ( essay , additional resources ) b. pyramid of khafre and the great sphinx ( essay , additional resources ) c. pyramid of menkaura ( essay , additional resources ) 18 . king menkaura and queen ( essay , images , additional resources ) 19 . the law code stele of hammurabi ( essay , video , images , additional resources ) 20 . temple of amun-re and hypostyle hall ( essay , video , images , additional resources ) 21 . mortuary temple of hatshepsut ( video , images , additional resources ) 22 . akhenaton , nefertiti , and three daughters ( video , images , additional resources ) 23 . tutankhamun 's tomb , innermost coffin ( essay , images , additional resources ) 24 . last judgment of hu-nefer , ( book of the dead ) ( video , essay , images , additional resources ) 25 . lamassu from the citadel of sargon ii , dur sharrukin ( modern iraq ) ( video , images , additional resources ) 26 . athenian agora ( video , images , additional resources ) 27 . anavysos kouros ( video , images , additional resources ) 28 . peplos kore from the acropolis ( video , images , additional resources ) 29 . sarcophagus of the spouses ( video , essay , images , additional resources ) 30 . audience hall ( apadana ) of darius and xeres ( essay , images , additional resources ) related : column capital , audience hall ( apadana ) of darius at susa ( video , images ) 31 . temple of minerva ( veii near rome , italy ) , sculpture of apollo ( essay , video , images , additional resources ) 32 . tomb of the triclinium ( essay , images , additional resources ) 33 . niobides krater ( video , images , additional resources ) 34 . doryphoros ( spear bearer ) ( video , essay , images , additional resources ) 35 . acropolis a. parthenon ( video , essay , images , additional resources ) i. helios , horses , dionysus ( heracles ? ) , east pediment ( video , images , additional resources ) ii . frieze ( video , images ) , plaque of the ergastines ( video , images , additional resources ) b . victory adjusting her sandel , temple of athena nike ( video , images , additional resources ) 36 . grave stele of hegeso ( video , images , additional resources ) 37 . winged victory of samothrace ( video , images , additional resources ) 38 . great altar of zeus and athena at pergamon ( video , images , additional resources 39 . house of the vettii ( essay , images , additional resources ) 40 . alexander mosaic from the house of the faun , pompeii ( video , images , additional resources ) 41 . seated boxer ( video , images , additional resources ) 42 . head of a roman patrician ( essay , images , additional resources ) related : veristic male portrait ( video , additional resources ) 43 . augustus of prima porta ( video , essay , images , additional resources ) 44 . colosseum ( flavian amphitheater ) ( video , images , additional resources 45 . forum of trajan ( essay , additional resources ) a . forum ( video , images , additional resources ) b . column ( video , images , additional resources ) c. markets ( video , images , additional resources ) 46 . pantheon ( video , images , additional resources ) 47 . ludovisi battle sarcophagus ( video , images , additional resources ) content area 3 early europe and colonial americas 200-1750 c.e . 48 . catacomb of priscilla ( greek chapel , orant , good shepherd frescos ) ( video , images , additional resources ) 49 . santa sabina ( video , essay , images , additional resources ) 50 . vienna genesis b. jacob wrestling the angel ( video , images , additional resources ) a. rebecca and eliezer at the well ( essay , images ) 51 . san vitale ( including justinian and theodora panels ) ( video , essay , images , additional resources ) 52 . hagia sophia ( video , essay , images , additional resources ) a. theotokos mosaic ( video , images , additional resources ) b. deësis mosaic ( video , images , additional resources ) c. hagia sophia as a mosque ( video , images , additional resources ) 53 . merovingian looped fibulae ( essay , images , additional resources additional resources 2 ) 54 . virgin ( theotokos ) and child between saints theodore and george ( essay , images , additional resources ) 55 . lindisfarne gospels , st. matthew , cross-carpet page ; st. luke incipit page ( essay , images , additional resources ) 56 . great mosque , córdoba , spain ( essay , images , additional resources ) 57 . pyxis of al-mughira ( essay , images , additional resources ) 58 . church of sainte-foy and reliquary ( essay , images , additional resources ) 59 . bayeux tapestry ( essay , images , additional resources ) 60 . chartres cathedral ( video , image , additional resources ) 61 . dedication page with blanche of castile and king louis ix of france , essay , image , additional resources ) and scenes from the apocalypse ( essay , image , additional resources ) —both from bibles moralisée ( moralized bibles ) 62 . röttgen pietà ( video , essay , images ) 63 . arena ( scrovegni ) chapel , including lamentation ( additional resources ) a . introduction ( video ) b. fresco cycle ( video ) c. lamentation ( video ) d. last judgment ( video ) 64 . golden haggadah ( essay , images , additional resources 1 , additional resources 2 ) 65 . alhambra ( essay , additional resources ) 66 . annunciation triptych ( merode altarpiece ) ( video , images , additional resources ) 67 . pazzi chapel , filipo brunellschi ( video , images , additional resources ) 68 . the arnolfini portrait , jan van eyck ( video , additional resources ) 69 . david , donatello ( video , essay , images , additional resources ) 70 . palazzo rucellai , leon battista alberti ( video , essay , additional resources ) 71 . madonna and child with two angels , fra filippo lippi ( video , image , additionalresources ) 72 . birth of venus , sandro botticelli ( video , image , additional resources ) 73 . last supper , leonardo da vinci ( video , essay , image , additional resources 75 . sistine chapel ceiling and altar wall frescos , michelangelo a. ceiling ( video , essay , study for sibyl video , additional resources ) b. altar wall ( video , additional resources ) 76 . school of athens , raphael ( video , essay , images , additional resources ) 77.isenheim altarpiece , matthias grünewald ( essay , images , additional resources ) 78 . entombment of christ , jacobo da pontormo ( video , image , , additional resources ) 79 . allegory of law and grace , lucas cranach the elder ( essay , images , additionalresources ) 80 . venus of urbino , titian ( video , image , additional resources ) 81 . frontispiece of the codex mendoza ( essay , images , additional resources 82 . il gesù , including triumph of the name of jesus ceiling fresco ( video , image 1 , image 2 , additional resources ) 83 . hunters in the snow , pieter bruegel the elder ( video , image , additional resources ) 84 . mosque of selim ii ( essay , images additional resources ) 85 . calling of saint matthew , caravaggio ( video , images , additional resources ) 86 . henri iv receives the portrait of marie de'medici , from the marie de'medici cycle , peter paul rubens ( video , essay , images , additional resources ) 87 . self-portrait with saskia , rembrandt van rijn ( essay , image , additional resources ) 88 . san carlo alle quattro fontane , francesco borromini ( video , images , additionalresources ) 89 . ecstasy of saint teresa , gian lorenzo bernini ( video , images , additional resources ) 90 . angel with arquebus , asiel timor dei , master of calamarca ( essay , image , additional resources ) 91 . las meninas , diego velazquez ( video , image , additional resources ) 92 . woman holding a balance , johannes vermeer ( video , image , additional resources ) 93 . the palace at versailles ( essay , images and maps , additional resources ) 94 . screen with siege of belgrade and hunting scene ( video , images , additional resources ) 95 . the virgin of guadalupe ( virgen de guadalupe ) _ , miguel gonzález ( video , essay , additional resources ) 96 . fruit and insects , rachel ruysch ( video , images , additional resources ) 97 . spaniard and indian produce a mestizo , attributed to juan rodríguez juárez ( essay , image , additional resources ) 98 . the tête à tête , from marriage a la mode , william hogarth ( video , images , additional resources ) content area 4 later europe and americas 1750-1980 c.e . 99 . portrait of sor juana inés de la cruz , miguel cabrera ( essay , [ additional resources ] [ http : //jps.library.utoronto.ca/index.php/emw/article/viewfile/18743/15678 ] ) 100 . a philosopher giving a lecture on the orrery , joseph wright of derby ( essay , image , additional resources ) 101 . the swing , jean-honoré fragonard ( video , images , additional resources ) 102 . monticello , thomas jefferson ( essay , images additional resources 1 , 2 ) 103 . the oath of the horatii , jacques-louis david ( video , essay , images , additionalresources ) 104 . george washington , jean-antoine houdon ( essay , image , additional resources 1 , 2 ) 105 . self-portrait , elisabeth louise vigée-lebrun ( essay , image , additional resources ) 106 . y no hai remedio ( and there 's nothing to be done ) , from los desastres de la guerra ( the disasters of war ) , plate 15 , francesco de goya ( essay , images , additional resources 107 . la grande odalisque , jean-auguste-dominique ingres ( video , essay , images , additional resources ) 108 . liberty leading the people , eugène delacroix ( video , essay , images , additionalresources ) 109 . view from mount holyoke , northampton , massachusetts , after athunderstorm—the oxbow , thomas cole ( video , essay , images , additional resources ) 110 . still life in studio , louis-jacques-mandé daguerre ( essay , image , additional resources ) 111 . slave ship ( slavers throwing overboard the dead and dying , typhoon coming on ) , joseph mallord william turner ( video , images , additional resources ) 112 . palace of westminster ( houses of parliament ) , charles barry , a.w.n . pugin ( video , images , additional resources ) 113 . the stonebreakers , gustave courbet ( essay , image , additional resources 1 , 2 ) 114 . nadar elevating photography to art , honoré daumier ( essay , image , additionalresources ) 115 . olympia , édouard manet ( video , image , additional resources ) 116 . the saint-lazare station , claude monet ( video , image , additional resources 117 . the horse in motion , eadweard muybridge ( essay , image , additional resources ) 118 . the valley of mexico from the hillside of santa isabel , josé maría velasco ( video , essay , photos ) 119 . the burghers of calais , auguste rodin ( essay , image , 3d image , additional resources ) 120 . the starry night , vincent van gogh ( essay , image , additional resources ) 121 . the coiffure , mary cassatt ( essay , image , additional resources ) 122 . the scream , edvard munch ( essay , image , additional resources ) 123 . where do we come from ? what are we ? where are we going ? , paul gauguin ( essay , images , additional resources ) 124 . carson , pirie , scott and company building , louis sullivan ( essay , images , additionalresources ) 125 . mont sainte-victoire , paul cézanne ( video , essay , image , additional resources 126 . les demoiselles d'avignon , pablo picasso ( video , images , additional resources 127 . the steerage , alfred stieglitz ( essay , image , additional resources ) 128 . the kiss , gustav klimt ( video , image , additional resources ) 129 . the kiss , constantin brancusi ( video , images , additional resources ) 130 . the portuguese , georges braque ( essay , images , additional resources ) 131 . the goldfish , henri matisse ( essay , image , additional resources ) 132 . improvisation 28 ( second version ) , vasily kandinsky ( video , photo , additional resources 133 . self-portrait as a soldier , ernst ludwig kirchner ( essay , image , additional resources 134 . memorial sheet of karl liebknecht , käthe kollwitz ( essay , image , additional resources ) 135 . villa savoye , le corbusier ( essay , images , additional resources ) 136 . composition with red , blue and yellow , piet mondrian ( essay , image , additional resources ) 137 . illustration from the results of the first five-year plan , varvarastepanova ( essay , image , additional resources ) 138 . object ( le déjeuner en fourrure ) , meret oppenheim ( essay , quiz , image , additional resources 139 . fallingwater , frank lloyd wright ( essay , plans and elevations , images , additionalresources ) 140 . the two fridas , frida kahlo ( essay , image , additional resources ) 141 . the migration of the negro , panel no . 49 , jacob lawrence ( video-short version , video-long version , photo , additional resources ) 142 . the jungle , wilfredo lam ( essay , image , additional resources ) 143 . dream of a sunday afternoon in alameda central park , diego rivera ( essay , image , additional resources ) 144 . fountain , marcel duchamp ( video , images , additional resources ) 145 . woman i , willem de kooning ( video , images , additional resources ) 146 . seagram building , ludwig mies van der rohe , philip johnson ( video , images , additional resources ) 147 . marilyn diptych , andy warhol ( essay , image , additional resources ) 148 . narcissus garden , yayoi kusama ( essay , image , additional resources ) 149 . the bay , helen frankenthaler ( essay , quiz , image , additional resources ) 150 . lipstick ( ascending ) on caterpillar tracks , claes oldenburg ( essay , photo , additional resoures ) 151 . spiral jetty , robert smithson ( video , images , additional resources ) 152 . house in new castle county , robert ventura , john rausch and denise scott brown ( essay , photos , additional resources ) content area 5 indigenous americas 1000 b.c.e.-1980 c.e . 153 . chavín de huántar ( essay , image , additional resources ) 154 . mesa verde cliff dwellings ( essay , video , photos , additional resources ) 155 . yaxchilán lintel 25 , structure 23 ( essay , essay on related lintel , photo , additional resources ) 156 . great serpent mound ( essay , photos , additional resources ) 157 . templo mayor , main aztec temple ( video , essay , images , additional resources ) a . the coyolxauhqui stone ( video , images , additional resources ) b . calendar stone ( video , images , additional resources ) c. olmec-style mask ( video , images , additional resources ) 158 . ruler 's feather headdress ( probably of moctezuma ii ) ( video , images , additionalresources , see page 12 ) 159 . city of cusco ( essay , video , photos , additional resources ) 160 . maize cobs ( essay , image , additional resources 1 , 2 ) 161 . city of machu picchu ( essay , video , photos , additional resources ) 162 . all-t'oqapu tunic ( essay , photo , additional resources ) 163 . bandolier bag ( essay , photo , additional resources ) 164 . transformation mask ( essay , image , additional resources ) 165 . painted elk hide , attributed to cotsiogo ( cadzi cody ) ( essay , image , additionalresources ) 166 . black-on-black ceramic vessel , maria martínez and julian martínez ( essay , image , additional resources ) content area 6 africa 1100-1980 c.e . 167 . conical tower and circular wall of great zimbabwe ( video , images , additional resources ) 168 . great mosque of djenné ( essay , video , images , additional resources ) 169 . wall plaque , from oba 's palace ( essay , images , additional resources ) 170 . sika dwa kofi ( golden stool ) ( video , image , additional resources ) 171 . ndop ( portrait figure ) of king mishe mishyaang mambul ( essay , image , additional resources ) 172 . nkisi n ’ kondi ( essay , image , additional resources ) 173 . female ( pwo ) mask ( video , images , additional resources ) 174 . portrait mask ( mblo ) ( essay , image , additional resources ) 175 . bundu mask ( video , image , additional resources ) 176 . ikenga ( shrine figure ) ( video , image , additional resources ) 177 . lukasa ( memory board ) ( essay , image , additional resources ) 178 . aka elephant mask ( video , photos , additional resources 179 . reliquary figure ( byeri ) ( video , photos , additional resources ) 180 . veranda post of enthroned king and senior wife ( opo ogoga ) ( related video , image 1 , image 2 , additional resources ) content area 7 west and central asia 500 b.c.e.-1980 c.e . 181 . petra , jordan : treasury and great temple ( additional resources ) a. nabataeans introduction ( essay ) b. petra and the treasury ( essay , images ) c. petra and the great temple ( essay , images ) d. unesco siq project ( video ) 182 . buddha , bamiyan ( essay , image , video , additional resources 1 , 2 ) 183 . the kaaba ( essay , images , additional resources ) 184 . jowo rinpoche , enshrined in the jokhang temple ( essay , images , additional resources ) 185 . dome of the rock ( essay , images , additional resources ) 186 . great mosque ( masjid-e jameh ) , isfahan ( essay , images , additional resources ) 187 . folio from a qur'an ( essay , image , additional resources ) 188 . basin ( baptistère de saint louis ) , mohammed ibn al-zain ( video , images , additional resources ) 189 . bahram gur fights the karg , folio from the great il-khanid shahnama ( essay , image , additional resources ) 190 . the court of gayumars , folio from shah tahmasp's shahnama ( essay , image , additional resources ) 191 . the ardabil carpet ( essay , images , additional resources ) content area 8 south , east and southeast asia 300 b.c.e.-1980 c.e . 192 . great stupa at sanchi ( essay , video , images , additional resources ) 193 . terracotta warriors from mausoleum of the first qin emperor of china ( essay , video , images , additional resources ) 194 . funeral banner of lady dai ( xin zhui ) ( essay , images , additional resources 195 . longmen caves ( essay , video , images , additional resources ) 196 . gold and jade crown ( essay , image , additional resources ) 197 . todai-ji ( essay , video , images , additional resources ) 198 . borobudur ( essay , images , additional resources ) 199 . angkor , the temple of angkor wat , the city of angkor thom , cambodia ( essay , video , images , additional resources ) 200 . lakshmana temple ( essay , additional resources ) 201 . travelers among mountains and streams , fan kuan ( essay , image , additional resources ) 202 . shiva as lord of dance ( nataraja ) ( essay , image , additional resources ) 203 . night attack on the sanjô palace ( essay , photos , additional resources ) 204 . the david vases ( video , essay , images , additional resources 1 , 2 ) 205 . portrait of sin sukju ( essay , image , additional resources ) 206 . forbidden city ( essay , video , images , additional resources ) 207 . ryoan-ji ( video , essay , images , additional resources 1 , 2 ) 208 . jahangir preferring a sufi shaikh to kings , bichitr ( essay , image , additional resources ) 209 . taj mahal ( essay , images , additional resources ) 210 . white and red plum blossoms , ogata korin ( essay , image , additional resources 1 , additional resources 2 , additional resources 3 ) 211 . under the wave off kanagawa ( kanagawa oki nami ura ) , also known as the great wave , from the series thirty-six views of mount fuji , katsushika hokusai ( essay , image , additional resources ) 212 . chairman mao en route to anyuan ( essay , image additional resources ) content area 9 the pacific 700-1980 c.e . 213 . nan madol 214 . moai on platform ( ahu ) ( video , essay , images , additional resources 1 , 2 , 3 ) 215 . 'ahu 'ula ( feather cape ) ( essay , image , additional resources ) 216 . staff god ( essay , images , additional resources ) 217 . female deity from nukuoro ( essay , images , additional resources 1 , 2 , 3 ) 218 . buk mask ( video , images , additional resources ) 219 . hiapo ( tapa ) ( essay , image , additional resources ) 220 . tamati waka nene , gottfried lindaur ( essay , image , additional resources 221 . navigation chart ( essay , image , additional resources ) 222 . malagan display and mask ( essay , image 1 , 2 , additional resources ) 223 . presentation of fijian mats and tapas cloths to queen elizabeth ii ( essay , image , additional resources 1 , 2 ) content area 10 global contemporary 1980 c.e . to present 224 . the gates , christo and jeanne-claude ( essay , images , additional resources ) 225 . vietnam veterans memorial , maya lin ( video , images , additional resources ) 226 . horn players , jean-michel basquiat ( essay , image , additional resources ) 227 . summer trees , song su-nam ( essay , image , additional resources ) 228 . androgyne iii , magdalena abakanowicz ( essay , images , additional resources ) 229 . a book from the sky , xu bing ( video , images , additional resources 230 . pink panther , jeff koons ( essay , images , additional resources ) 231 . untitled ( # 228 ) , from the history portraits series , cindy sherman ( essay , image , additional resources ) 232 . dancing at the louvre , from the series , the french collection , part 1 ; # 1 , faith ringgold ( essay , image , additional resources ) 233 . trade ( gifts for trading land with white people ) , jaune quick-to-see smith ( essay , image , additional resources ) 234 . earth ’ s creation , emily kame kngwarreye ( essay , image , additional resources ) 235 . rebellious silence , from the women of allah series , shirin neshat ( artist ) ; photo by cynthia preston ( essay , image , additional resources ) 236 . en la barberia no se llora ( no crying allowed in the barbershop ) , pepon osorio ( essay , image 1 , 2 , additional resources ) 237 . pisupo lua afe ( corned beef 2000 ) , michel tuffery ( essay , artist , biography , video 1 , video 2 , images ) 238 . electronic superhighway , nam june paik ( essay , image , additional resources 239 . the crossing , bill viola ( essay , image , additional resources ) 240 . guggenheim museum bilbao , frank gehry ( essay , images , additional resources ) 241 . pure land , mariko mori ( essay , image , additional resources ) 242 . lying with the wolf , kiki smith ( essay , image , additional resources ) 243.darkytown rebellion , kara walker ( essay , image , additional resources ) 244 . the swing ( after fragonard ) , yinka shonibare ( essay , images , additional resources ) 245 . old man ’ s cloth , el anatsui ( video , essay , image , additional resources ) 246 . stadia ii , julie mehretu ( essay , image , additional resources ) 247 . preying mantra , wangechi mutu ( essay , image , additional resources ) 248 . shibboleth , doris salcedo ( essay , images , additional resources ) 249 . maxxi national museum of xxi century arts , zaha hadid ( video , images , additional resources ) 250 . kui hua zi ( sunflower seeds ) , ai weiwei ( essay , video , images , additional resources ) special thanks to the many art historians and curators who have contributed their expertise as well as our museum partners , the american museum of natural history , the asian art museum , the british museum , the j. paul getty museum , the metropolitan museum of art , the museum of modern art , and tate . ap art history is a registered trademark of the college board , which was not involved in the production of , and does not endorse , this product .
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this list links to videos , essays , images , and additional resources for the 250 required works of art for the ap* art history course and exam . smarthistory is adding new materials regularly .
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is there a website where i can take a mock multiple choice of ap art history ?
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this list links to videos , essays , images , and additional resources for the 250 required works of art for the ap* art history course and exam . smarthistory is adding new materials regularly . for more context on these works of art , please see the art history sections of khan academy . content area 1 global prehistory 30,000-500b.c.e . 1 . apollo 11 stones ( essay , image , additional resources ) 2 . great hall of bulls ( essay , image , additional resources ) 3 . camelid sacrum in the shape of a canine ( essay , image , additional resources ) 4 . running horned woman ( essay , image , additional resources ) 5 . bushel with ibex motifs ( video , images , additional resources ) 6 . anthropomorphic stele ( essay , image , additional resources ) 7 . jade cong ( video , article , image , additional resources ) 8 . stonehenge ( essay , video , image 1 , image 2 , additional resources ) 9 . ambum stone ( essay , image , additional resources ) 10 . tlatilco female figurine ( video , essay , image , additional resources ) 11 . terracotta fragment ( lapita ) ( essay , image , additional resources ) content area 2 ancient mediterranean 3,500-300 b.c.e . 12 . white temple and its ziggurat ( essay , image 1 , image 2 additional resources ) 13 . palette of king narmer ( essay , images , additional resources ) 14 . statues of votive figures , from the square temple at eshnunna ( video , images , additional resources ) 15 . seated scribe ( video , images , additional resources ) 16 . standard of ur from the royal tombs at ur ( video , essay , images , additional resources 17 . great pyramids of giza ( essay , images , additional resources ) a. pyramid of khufu ( essay , additional resources ) b. pyramid of khafre and the great sphinx ( essay , additional resources ) c. pyramid of menkaura ( essay , additional resources ) 18 . king menkaura and queen ( essay , images , additional resources ) 19 . the law code stele of hammurabi ( essay , video , images , additional resources ) 20 . temple of amun-re and hypostyle hall ( essay , video , images , additional resources ) 21 . mortuary temple of hatshepsut ( video , images , additional resources ) 22 . akhenaton , nefertiti , and three daughters ( video , images , additional resources ) 23 . tutankhamun 's tomb , innermost coffin ( essay , images , additional resources ) 24 . last judgment of hu-nefer , ( book of the dead ) ( video , essay , images , additional resources ) 25 . lamassu from the citadel of sargon ii , dur sharrukin ( modern iraq ) ( video , images , additional resources ) 26 . athenian agora ( video , images , additional resources ) 27 . anavysos kouros ( video , images , additional resources ) 28 . peplos kore from the acropolis ( video , images , additional resources ) 29 . sarcophagus of the spouses ( video , essay , images , additional resources ) 30 . audience hall ( apadana ) of darius and xeres ( essay , images , additional resources ) related : column capital , audience hall ( apadana ) of darius at susa ( video , images ) 31 . temple of minerva ( veii near rome , italy ) , sculpture of apollo ( essay , video , images , additional resources ) 32 . tomb of the triclinium ( essay , images , additional resources ) 33 . niobides krater ( video , images , additional resources ) 34 . doryphoros ( spear bearer ) ( video , essay , images , additional resources ) 35 . acropolis a. parthenon ( video , essay , images , additional resources ) i. helios , horses , dionysus ( heracles ? ) , east pediment ( video , images , additional resources ) ii . frieze ( video , images ) , plaque of the ergastines ( video , images , additional resources ) b . victory adjusting her sandel , temple of athena nike ( video , images , additional resources ) 36 . grave stele of hegeso ( video , images , additional resources ) 37 . winged victory of samothrace ( video , images , additional resources ) 38 . great altar of zeus and athena at pergamon ( video , images , additional resources 39 . house of the vettii ( essay , images , additional resources ) 40 . alexander mosaic from the house of the faun , pompeii ( video , images , additional resources ) 41 . seated boxer ( video , images , additional resources ) 42 . head of a roman patrician ( essay , images , additional resources ) related : veristic male portrait ( video , additional resources ) 43 . augustus of prima porta ( video , essay , images , additional resources ) 44 . colosseum ( flavian amphitheater ) ( video , images , additional resources 45 . forum of trajan ( essay , additional resources ) a . forum ( video , images , additional resources ) b . column ( video , images , additional resources ) c. markets ( video , images , additional resources ) 46 . pantheon ( video , images , additional resources ) 47 . ludovisi battle sarcophagus ( video , images , additional resources ) content area 3 early europe and colonial americas 200-1750 c.e . 48 . catacomb of priscilla ( greek chapel , orant , good shepherd frescos ) ( video , images , additional resources ) 49 . santa sabina ( video , essay , images , additional resources ) 50 . vienna genesis b. jacob wrestling the angel ( video , images , additional resources ) a. rebecca and eliezer at the well ( essay , images ) 51 . san vitale ( including justinian and theodora panels ) ( video , essay , images , additional resources ) 52 . hagia sophia ( video , essay , images , additional resources ) a. theotokos mosaic ( video , images , additional resources ) b. deësis mosaic ( video , images , additional resources ) c. hagia sophia as a mosque ( video , images , additional resources ) 53 . merovingian looped fibulae ( essay , images , additional resources additional resources 2 ) 54 . virgin ( theotokos ) and child between saints theodore and george ( essay , images , additional resources ) 55 . lindisfarne gospels , st. matthew , cross-carpet page ; st. luke incipit page ( essay , images , additional resources ) 56 . great mosque , córdoba , spain ( essay , images , additional resources ) 57 . pyxis of al-mughira ( essay , images , additional resources ) 58 . church of sainte-foy and reliquary ( essay , images , additional resources ) 59 . bayeux tapestry ( essay , images , additional resources ) 60 . chartres cathedral ( video , image , additional resources ) 61 . dedication page with blanche of castile and king louis ix of france , essay , image , additional resources ) and scenes from the apocalypse ( essay , image , additional resources ) —both from bibles moralisée ( moralized bibles ) 62 . röttgen pietà ( video , essay , images ) 63 . arena ( scrovegni ) chapel , including lamentation ( additional resources ) a . introduction ( video ) b. fresco cycle ( video ) c. lamentation ( video ) d. last judgment ( video ) 64 . golden haggadah ( essay , images , additional resources 1 , additional resources 2 ) 65 . alhambra ( essay , additional resources ) 66 . annunciation triptych ( merode altarpiece ) ( video , images , additional resources ) 67 . pazzi chapel , filipo brunellschi ( video , images , additional resources ) 68 . the arnolfini portrait , jan van eyck ( video , additional resources ) 69 . david , donatello ( video , essay , images , additional resources ) 70 . palazzo rucellai , leon battista alberti ( video , essay , additional resources ) 71 . madonna and child with two angels , fra filippo lippi ( video , image , additionalresources ) 72 . birth of venus , sandro botticelli ( video , image , additional resources ) 73 . last supper , leonardo da vinci ( video , essay , image , additional resources 75 . sistine chapel ceiling and altar wall frescos , michelangelo a. ceiling ( video , essay , study for sibyl video , additional resources ) b. altar wall ( video , additional resources ) 76 . school of athens , raphael ( video , essay , images , additional resources ) 77.isenheim altarpiece , matthias grünewald ( essay , images , additional resources ) 78 . entombment of christ , jacobo da pontormo ( video , image , , additional resources ) 79 . allegory of law and grace , lucas cranach the elder ( essay , images , additionalresources ) 80 . venus of urbino , titian ( video , image , additional resources ) 81 . frontispiece of the codex mendoza ( essay , images , additional resources 82 . il gesù , including triumph of the name of jesus ceiling fresco ( video , image 1 , image 2 , additional resources ) 83 . hunters in the snow , pieter bruegel the elder ( video , image , additional resources ) 84 . mosque of selim ii ( essay , images additional resources ) 85 . calling of saint matthew , caravaggio ( video , images , additional resources ) 86 . henri iv receives the portrait of marie de'medici , from the marie de'medici cycle , peter paul rubens ( video , essay , images , additional resources ) 87 . self-portrait with saskia , rembrandt van rijn ( essay , image , additional resources ) 88 . san carlo alle quattro fontane , francesco borromini ( video , images , additionalresources ) 89 . ecstasy of saint teresa , gian lorenzo bernini ( video , images , additional resources ) 90 . angel with arquebus , asiel timor dei , master of calamarca ( essay , image , additional resources ) 91 . las meninas , diego velazquez ( video , image , additional resources ) 92 . woman holding a balance , johannes vermeer ( video , image , additional resources ) 93 . the palace at versailles ( essay , images and maps , additional resources ) 94 . screen with siege of belgrade and hunting scene ( video , images , additional resources ) 95 . the virgin of guadalupe ( virgen de guadalupe ) _ , miguel gonzález ( video , essay , additional resources ) 96 . fruit and insects , rachel ruysch ( video , images , additional resources ) 97 . spaniard and indian produce a mestizo , attributed to juan rodríguez juárez ( essay , image , additional resources ) 98 . the tête à tête , from marriage a la mode , william hogarth ( video , images , additional resources ) content area 4 later europe and americas 1750-1980 c.e . 99 . portrait of sor juana inés de la cruz , miguel cabrera ( essay , [ additional resources ] [ http : //jps.library.utoronto.ca/index.php/emw/article/viewfile/18743/15678 ] ) 100 . a philosopher giving a lecture on the orrery , joseph wright of derby ( essay , image , additional resources ) 101 . the swing , jean-honoré fragonard ( video , images , additional resources ) 102 . monticello , thomas jefferson ( essay , images additional resources 1 , 2 ) 103 . the oath of the horatii , jacques-louis david ( video , essay , images , additionalresources ) 104 . george washington , jean-antoine houdon ( essay , image , additional resources 1 , 2 ) 105 . self-portrait , elisabeth louise vigée-lebrun ( essay , image , additional resources ) 106 . y no hai remedio ( and there 's nothing to be done ) , from los desastres de la guerra ( the disasters of war ) , plate 15 , francesco de goya ( essay , images , additional resources 107 . la grande odalisque , jean-auguste-dominique ingres ( video , essay , images , additional resources ) 108 . liberty leading the people , eugène delacroix ( video , essay , images , additionalresources ) 109 . view from mount holyoke , northampton , massachusetts , after athunderstorm—the oxbow , thomas cole ( video , essay , images , additional resources ) 110 . still life in studio , louis-jacques-mandé daguerre ( essay , image , additional resources ) 111 . slave ship ( slavers throwing overboard the dead and dying , typhoon coming on ) , joseph mallord william turner ( video , images , additional resources ) 112 . palace of westminster ( houses of parliament ) , charles barry , a.w.n . pugin ( video , images , additional resources ) 113 . the stonebreakers , gustave courbet ( essay , image , additional resources 1 , 2 ) 114 . nadar elevating photography to art , honoré daumier ( essay , image , additionalresources ) 115 . olympia , édouard manet ( video , image , additional resources ) 116 . the saint-lazare station , claude monet ( video , image , additional resources 117 . the horse in motion , eadweard muybridge ( essay , image , additional resources ) 118 . the valley of mexico from the hillside of santa isabel , josé maría velasco ( video , essay , photos ) 119 . the burghers of calais , auguste rodin ( essay , image , 3d image , additional resources ) 120 . the starry night , vincent van gogh ( essay , image , additional resources ) 121 . the coiffure , mary cassatt ( essay , image , additional resources ) 122 . the scream , edvard munch ( essay , image , additional resources ) 123 . where do we come from ? what are we ? where are we going ? , paul gauguin ( essay , images , additional resources ) 124 . carson , pirie , scott and company building , louis sullivan ( essay , images , additionalresources ) 125 . mont sainte-victoire , paul cézanne ( video , essay , image , additional resources 126 . les demoiselles d'avignon , pablo picasso ( video , images , additional resources 127 . the steerage , alfred stieglitz ( essay , image , additional resources ) 128 . the kiss , gustav klimt ( video , image , additional resources ) 129 . the kiss , constantin brancusi ( video , images , additional resources ) 130 . the portuguese , georges braque ( essay , images , additional resources ) 131 . the goldfish , henri matisse ( essay , image , additional resources ) 132 . improvisation 28 ( second version ) , vasily kandinsky ( video , photo , additional resources 133 . self-portrait as a soldier , ernst ludwig kirchner ( essay , image , additional resources 134 . memorial sheet of karl liebknecht , käthe kollwitz ( essay , image , additional resources ) 135 . villa savoye , le corbusier ( essay , images , additional resources ) 136 . composition with red , blue and yellow , piet mondrian ( essay , image , additional resources ) 137 . illustration from the results of the first five-year plan , varvarastepanova ( essay , image , additional resources ) 138 . object ( le déjeuner en fourrure ) , meret oppenheim ( essay , quiz , image , additional resources 139 . fallingwater , frank lloyd wright ( essay , plans and elevations , images , additionalresources ) 140 . the two fridas , frida kahlo ( essay , image , additional resources ) 141 . the migration of the negro , panel no . 49 , jacob lawrence ( video-short version , video-long version , photo , additional resources ) 142 . the jungle , wilfredo lam ( essay , image , additional resources ) 143 . dream of a sunday afternoon in alameda central park , diego rivera ( essay , image , additional resources ) 144 . fountain , marcel duchamp ( video , images , additional resources ) 145 . woman i , willem de kooning ( video , images , additional resources ) 146 . seagram building , ludwig mies van der rohe , philip johnson ( video , images , additional resources ) 147 . marilyn diptych , andy warhol ( essay , image , additional resources ) 148 . narcissus garden , yayoi kusama ( essay , image , additional resources ) 149 . the bay , helen frankenthaler ( essay , quiz , image , additional resources ) 150 . lipstick ( ascending ) on caterpillar tracks , claes oldenburg ( essay , photo , additional resoures ) 151 . spiral jetty , robert smithson ( video , images , additional resources ) 152 . house in new castle county , robert ventura , john rausch and denise scott brown ( essay , photos , additional resources ) content area 5 indigenous americas 1000 b.c.e.-1980 c.e . 153 . chavín de huántar ( essay , image , additional resources ) 154 . mesa verde cliff dwellings ( essay , video , photos , additional resources ) 155 . yaxchilán lintel 25 , structure 23 ( essay , essay on related lintel , photo , additional resources ) 156 . great serpent mound ( essay , photos , additional resources ) 157 . templo mayor , main aztec temple ( video , essay , images , additional resources ) a . the coyolxauhqui stone ( video , images , additional resources ) b . calendar stone ( video , images , additional resources ) c. olmec-style mask ( video , images , additional resources ) 158 . ruler 's feather headdress ( probably of moctezuma ii ) ( video , images , additionalresources , see page 12 ) 159 . city of cusco ( essay , video , photos , additional resources ) 160 . maize cobs ( essay , image , additional resources 1 , 2 ) 161 . city of machu picchu ( essay , video , photos , additional resources ) 162 . all-t'oqapu tunic ( essay , photo , additional resources ) 163 . bandolier bag ( essay , photo , additional resources ) 164 . transformation mask ( essay , image , additional resources ) 165 . painted elk hide , attributed to cotsiogo ( cadzi cody ) ( essay , image , additionalresources ) 166 . black-on-black ceramic vessel , maria martínez and julian martínez ( essay , image , additional resources ) content area 6 africa 1100-1980 c.e . 167 . conical tower and circular wall of great zimbabwe ( video , images , additional resources ) 168 . great mosque of djenné ( essay , video , images , additional resources ) 169 . wall plaque , from oba 's palace ( essay , images , additional resources ) 170 . sika dwa kofi ( golden stool ) ( video , image , additional resources ) 171 . ndop ( portrait figure ) of king mishe mishyaang mambul ( essay , image , additional resources ) 172 . nkisi n ’ kondi ( essay , image , additional resources ) 173 . female ( pwo ) mask ( video , images , additional resources ) 174 . portrait mask ( mblo ) ( essay , image , additional resources ) 175 . bundu mask ( video , image , additional resources ) 176 . ikenga ( shrine figure ) ( video , image , additional resources ) 177 . lukasa ( memory board ) ( essay , image , additional resources ) 178 . aka elephant mask ( video , photos , additional resources 179 . reliquary figure ( byeri ) ( video , photos , additional resources ) 180 . veranda post of enthroned king and senior wife ( opo ogoga ) ( related video , image 1 , image 2 , additional resources ) content area 7 west and central asia 500 b.c.e.-1980 c.e . 181 . petra , jordan : treasury and great temple ( additional resources ) a. nabataeans introduction ( essay ) b. petra and the treasury ( essay , images ) c. petra and the great temple ( essay , images ) d. unesco siq project ( video ) 182 . buddha , bamiyan ( essay , image , video , additional resources 1 , 2 ) 183 . the kaaba ( essay , images , additional resources ) 184 . jowo rinpoche , enshrined in the jokhang temple ( essay , images , additional resources ) 185 . dome of the rock ( essay , images , additional resources ) 186 . great mosque ( masjid-e jameh ) , isfahan ( essay , images , additional resources ) 187 . folio from a qur'an ( essay , image , additional resources ) 188 . basin ( baptistère de saint louis ) , mohammed ibn al-zain ( video , images , additional resources ) 189 . bahram gur fights the karg , folio from the great il-khanid shahnama ( essay , image , additional resources ) 190 . the court of gayumars , folio from shah tahmasp's shahnama ( essay , image , additional resources ) 191 . the ardabil carpet ( essay , images , additional resources ) content area 8 south , east and southeast asia 300 b.c.e.-1980 c.e . 192 . great stupa at sanchi ( essay , video , images , additional resources ) 193 . terracotta warriors from mausoleum of the first qin emperor of china ( essay , video , images , additional resources ) 194 . funeral banner of lady dai ( xin zhui ) ( essay , images , additional resources 195 . longmen caves ( essay , video , images , additional resources ) 196 . gold and jade crown ( essay , image , additional resources ) 197 . todai-ji ( essay , video , images , additional resources ) 198 . borobudur ( essay , images , additional resources ) 199 . angkor , the temple of angkor wat , the city of angkor thom , cambodia ( essay , video , images , additional resources ) 200 . lakshmana temple ( essay , additional resources ) 201 . travelers among mountains and streams , fan kuan ( essay , image , additional resources ) 202 . shiva as lord of dance ( nataraja ) ( essay , image , additional resources ) 203 . night attack on the sanjô palace ( essay , photos , additional resources ) 204 . the david vases ( video , essay , images , additional resources 1 , 2 ) 205 . portrait of sin sukju ( essay , image , additional resources ) 206 . forbidden city ( essay , video , images , additional resources ) 207 . ryoan-ji ( video , essay , images , additional resources 1 , 2 ) 208 . jahangir preferring a sufi shaikh to kings , bichitr ( essay , image , additional resources ) 209 . taj mahal ( essay , images , additional resources ) 210 . white and red plum blossoms , ogata korin ( essay , image , additional resources 1 , additional resources 2 , additional resources 3 ) 211 . under the wave off kanagawa ( kanagawa oki nami ura ) , also known as the great wave , from the series thirty-six views of mount fuji , katsushika hokusai ( essay , image , additional resources ) 212 . chairman mao en route to anyuan ( essay , image additional resources ) content area 9 the pacific 700-1980 c.e . 213 . nan madol 214 . moai on platform ( ahu ) ( video , essay , images , additional resources 1 , 2 , 3 ) 215 . 'ahu 'ula ( feather cape ) ( essay , image , additional resources ) 216 . staff god ( essay , images , additional resources ) 217 . female deity from nukuoro ( essay , images , additional resources 1 , 2 , 3 ) 218 . buk mask ( video , images , additional resources ) 219 . hiapo ( tapa ) ( essay , image , additional resources ) 220 . tamati waka nene , gottfried lindaur ( essay , image , additional resources 221 . navigation chart ( essay , image , additional resources ) 222 . malagan display and mask ( essay , image 1 , 2 , additional resources ) 223 . presentation of fijian mats and tapas cloths to queen elizabeth ii ( essay , image , additional resources 1 , 2 ) content area 10 global contemporary 1980 c.e . to present 224 . the gates , christo and jeanne-claude ( essay , images , additional resources ) 225 . vietnam veterans memorial , maya lin ( video , images , additional resources ) 226 . horn players , jean-michel basquiat ( essay , image , additional resources ) 227 . summer trees , song su-nam ( essay , image , additional resources ) 228 . androgyne iii , magdalena abakanowicz ( essay , images , additional resources ) 229 . a book from the sky , xu bing ( video , images , additional resources 230 . pink panther , jeff koons ( essay , images , additional resources ) 231 . untitled ( # 228 ) , from the history portraits series , cindy sherman ( essay , image , additional resources ) 232 . dancing at the louvre , from the series , the french collection , part 1 ; # 1 , faith ringgold ( essay , image , additional resources ) 233 . trade ( gifts for trading land with white people ) , jaune quick-to-see smith ( essay , image , additional resources ) 234 . earth ’ s creation , emily kame kngwarreye ( essay , image , additional resources ) 235 . rebellious silence , from the women of allah series , shirin neshat ( artist ) ; photo by cynthia preston ( essay , image , additional resources ) 236 . en la barberia no se llora ( no crying allowed in the barbershop ) , pepon osorio ( essay , image 1 , 2 , additional resources ) 237 . pisupo lua afe ( corned beef 2000 ) , michel tuffery ( essay , artist , biography , video 1 , video 2 , images ) 238 . electronic superhighway , nam june paik ( essay , image , additional resources 239 . the crossing , bill viola ( essay , image , additional resources ) 240 . guggenheim museum bilbao , frank gehry ( essay , images , additional resources ) 241 . pure land , mariko mori ( essay , image , additional resources ) 242 . lying with the wolf , kiki smith ( essay , image , additional resources ) 243.darkytown rebellion , kara walker ( essay , image , additional resources ) 244 . the swing ( after fragonard ) , yinka shonibare ( essay , images , additional resources ) 245 . old man ’ s cloth , el anatsui ( video , essay , image , additional resources ) 246 . stadia ii , julie mehretu ( essay , image , additional resources ) 247 . preying mantra , wangechi mutu ( essay , image , additional resources ) 248 . shibboleth , doris salcedo ( essay , images , additional resources ) 249 . maxxi national museum of xxi century arts , zaha hadid ( video , images , additional resources ) 250 . kui hua zi ( sunflower seeds ) , ai weiwei ( essay , video , images , additional resources ) special thanks to the many art historians and curators who have contributed their expertise as well as our museum partners , the american museum of natural history , the asian art museum , the british museum , the j. paul getty museum , the metropolitan museum of art , the museum of modern art , and tate . ap art history is a registered trademark of the college board , which was not involved in the production of , and does not endorse , this product .
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smarthistory is adding new materials regularly . for more context on these works of art , please see the art history sections of khan academy . content area 1 global prehistory 30,000-500b.c.e .
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do you think children must be denied access to such art ?
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this list links to videos , essays , images , and additional resources for the 250 required works of art for the ap* art history course and exam . smarthistory is adding new materials regularly . for more context on these works of art , please see the art history sections of khan academy . content area 1 global prehistory 30,000-500b.c.e . 1 . apollo 11 stones ( essay , image , additional resources ) 2 . great hall of bulls ( essay , image , additional resources ) 3 . camelid sacrum in the shape of a canine ( essay , image , additional resources ) 4 . running horned woman ( essay , image , additional resources ) 5 . bushel with ibex motifs ( video , images , additional resources ) 6 . anthropomorphic stele ( essay , image , additional resources ) 7 . jade cong ( video , article , image , additional resources ) 8 . stonehenge ( essay , video , image 1 , image 2 , additional resources ) 9 . ambum stone ( essay , image , additional resources ) 10 . tlatilco female figurine ( video , essay , image , additional resources ) 11 . terracotta fragment ( lapita ) ( essay , image , additional resources ) content area 2 ancient mediterranean 3,500-300 b.c.e . 12 . white temple and its ziggurat ( essay , image 1 , image 2 additional resources ) 13 . palette of king narmer ( essay , images , additional resources ) 14 . statues of votive figures , from the square temple at eshnunna ( video , images , additional resources ) 15 . seated scribe ( video , images , additional resources ) 16 . standard of ur from the royal tombs at ur ( video , essay , images , additional resources 17 . great pyramids of giza ( essay , images , additional resources ) a. pyramid of khufu ( essay , additional resources ) b. pyramid of khafre and the great sphinx ( essay , additional resources ) c. pyramid of menkaura ( essay , additional resources ) 18 . king menkaura and queen ( essay , images , additional resources ) 19 . the law code stele of hammurabi ( essay , video , images , additional resources ) 20 . temple of amun-re and hypostyle hall ( essay , video , images , additional resources ) 21 . mortuary temple of hatshepsut ( video , images , additional resources ) 22 . akhenaton , nefertiti , and three daughters ( video , images , additional resources ) 23 . tutankhamun 's tomb , innermost coffin ( essay , images , additional resources ) 24 . last judgment of hu-nefer , ( book of the dead ) ( video , essay , images , additional resources ) 25 . lamassu from the citadel of sargon ii , dur sharrukin ( modern iraq ) ( video , images , additional resources ) 26 . athenian agora ( video , images , additional resources ) 27 . anavysos kouros ( video , images , additional resources ) 28 . peplos kore from the acropolis ( video , images , additional resources ) 29 . sarcophagus of the spouses ( video , essay , images , additional resources ) 30 . audience hall ( apadana ) of darius and xeres ( essay , images , additional resources ) related : column capital , audience hall ( apadana ) of darius at susa ( video , images ) 31 . temple of minerva ( veii near rome , italy ) , sculpture of apollo ( essay , video , images , additional resources ) 32 . tomb of the triclinium ( essay , images , additional resources ) 33 . niobides krater ( video , images , additional resources ) 34 . doryphoros ( spear bearer ) ( video , essay , images , additional resources ) 35 . acropolis a. parthenon ( video , essay , images , additional resources ) i. helios , horses , dionysus ( heracles ? ) , east pediment ( video , images , additional resources ) ii . frieze ( video , images ) , plaque of the ergastines ( video , images , additional resources ) b . victory adjusting her sandel , temple of athena nike ( video , images , additional resources ) 36 . grave stele of hegeso ( video , images , additional resources ) 37 . winged victory of samothrace ( video , images , additional resources ) 38 . great altar of zeus and athena at pergamon ( video , images , additional resources 39 . house of the vettii ( essay , images , additional resources ) 40 . alexander mosaic from the house of the faun , pompeii ( video , images , additional resources ) 41 . seated boxer ( video , images , additional resources ) 42 . head of a roman patrician ( essay , images , additional resources ) related : veristic male portrait ( video , additional resources ) 43 . augustus of prima porta ( video , essay , images , additional resources ) 44 . colosseum ( flavian amphitheater ) ( video , images , additional resources 45 . forum of trajan ( essay , additional resources ) a . forum ( video , images , additional resources ) b . column ( video , images , additional resources ) c. markets ( video , images , additional resources ) 46 . pantheon ( video , images , additional resources ) 47 . ludovisi battle sarcophagus ( video , images , additional resources ) content area 3 early europe and colonial americas 200-1750 c.e . 48 . catacomb of priscilla ( greek chapel , orant , good shepherd frescos ) ( video , images , additional resources ) 49 . santa sabina ( video , essay , images , additional resources ) 50 . vienna genesis b. jacob wrestling the angel ( video , images , additional resources ) a. rebecca and eliezer at the well ( essay , images ) 51 . san vitale ( including justinian and theodora panels ) ( video , essay , images , additional resources ) 52 . hagia sophia ( video , essay , images , additional resources ) a. theotokos mosaic ( video , images , additional resources ) b. deësis mosaic ( video , images , additional resources ) c. hagia sophia as a mosque ( video , images , additional resources ) 53 . merovingian looped fibulae ( essay , images , additional resources additional resources 2 ) 54 . virgin ( theotokos ) and child between saints theodore and george ( essay , images , additional resources ) 55 . lindisfarne gospels , st. matthew , cross-carpet page ; st. luke incipit page ( essay , images , additional resources ) 56 . great mosque , córdoba , spain ( essay , images , additional resources ) 57 . pyxis of al-mughira ( essay , images , additional resources ) 58 . church of sainte-foy and reliquary ( essay , images , additional resources ) 59 . bayeux tapestry ( essay , images , additional resources ) 60 . chartres cathedral ( video , image , additional resources ) 61 . dedication page with blanche of castile and king louis ix of france , essay , image , additional resources ) and scenes from the apocalypse ( essay , image , additional resources ) —both from bibles moralisée ( moralized bibles ) 62 . röttgen pietà ( video , essay , images ) 63 . arena ( scrovegni ) chapel , including lamentation ( additional resources ) a . introduction ( video ) b. fresco cycle ( video ) c. lamentation ( video ) d. last judgment ( video ) 64 . golden haggadah ( essay , images , additional resources 1 , additional resources 2 ) 65 . alhambra ( essay , additional resources ) 66 . annunciation triptych ( merode altarpiece ) ( video , images , additional resources ) 67 . pazzi chapel , filipo brunellschi ( video , images , additional resources ) 68 . the arnolfini portrait , jan van eyck ( video , additional resources ) 69 . david , donatello ( video , essay , images , additional resources ) 70 . palazzo rucellai , leon battista alberti ( video , essay , additional resources ) 71 . madonna and child with two angels , fra filippo lippi ( video , image , additionalresources ) 72 . birth of venus , sandro botticelli ( video , image , additional resources ) 73 . last supper , leonardo da vinci ( video , essay , image , additional resources 75 . sistine chapel ceiling and altar wall frescos , michelangelo a. ceiling ( video , essay , study for sibyl video , additional resources ) b. altar wall ( video , additional resources ) 76 . school of athens , raphael ( video , essay , images , additional resources ) 77.isenheim altarpiece , matthias grünewald ( essay , images , additional resources ) 78 . entombment of christ , jacobo da pontormo ( video , image , , additional resources ) 79 . allegory of law and grace , lucas cranach the elder ( essay , images , additionalresources ) 80 . venus of urbino , titian ( video , image , additional resources ) 81 . frontispiece of the codex mendoza ( essay , images , additional resources 82 . il gesù , including triumph of the name of jesus ceiling fresco ( video , image 1 , image 2 , additional resources ) 83 . hunters in the snow , pieter bruegel the elder ( video , image , additional resources ) 84 . mosque of selim ii ( essay , images additional resources ) 85 . calling of saint matthew , caravaggio ( video , images , additional resources ) 86 . henri iv receives the portrait of marie de'medici , from the marie de'medici cycle , peter paul rubens ( video , essay , images , additional resources ) 87 . self-portrait with saskia , rembrandt van rijn ( essay , image , additional resources ) 88 . san carlo alle quattro fontane , francesco borromini ( video , images , additionalresources ) 89 . ecstasy of saint teresa , gian lorenzo bernini ( video , images , additional resources ) 90 . angel with arquebus , asiel timor dei , master of calamarca ( essay , image , additional resources ) 91 . las meninas , diego velazquez ( video , image , additional resources ) 92 . woman holding a balance , johannes vermeer ( video , image , additional resources ) 93 . the palace at versailles ( essay , images and maps , additional resources ) 94 . screen with siege of belgrade and hunting scene ( video , images , additional resources ) 95 . the virgin of guadalupe ( virgen de guadalupe ) _ , miguel gonzález ( video , essay , additional resources ) 96 . fruit and insects , rachel ruysch ( video , images , additional resources ) 97 . spaniard and indian produce a mestizo , attributed to juan rodríguez juárez ( essay , image , additional resources ) 98 . the tête à tête , from marriage a la mode , william hogarth ( video , images , additional resources ) content area 4 later europe and americas 1750-1980 c.e . 99 . portrait of sor juana inés de la cruz , miguel cabrera ( essay , [ additional resources ] [ http : //jps.library.utoronto.ca/index.php/emw/article/viewfile/18743/15678 ] ) 100 . a philosopher giving a lecture on the orrery , joseph wright of derby ( essay , image , additional resources ) 101 . the swing , jean-honoré fragonard ( video , images , additional resources ) 102 . monticello , thomas jefferson ( essay , images additional resources 1 , 2 ) 103 . the oath of the horatii , jacques-louis david ( video , essay , images , additionalresources ) 104 . george washington , jean-antoine houdon ( essay , image , additional resources 1 , 2 ) 105 . self-portrait , elisabeth louise vigée-lebrun ( essay , image , additional resources ) 106 . y no hai remedio ( and there 's nothing to be done ) , from los desastres de la guerra ( the disasters of war ) , plate 15 , francesco de goya ( essay , images , additional resources 107 . la grande odalisque , jean-auguste-dominique ingres ( video , essay , images , additional resources ) 108 . liberty leading the people , eugène delacroix ( video , essay , images , additionalresources ) 109 . view from mount holyoke , northampton , massachusetts , after athunderstorm—the oxbow , thomas cole ( video , essay , images , additional resources ) 110 . still life in studio , louis-jacques-mandé daguerre ( essay , image , additional resources ) 111 . slave ship ( slavers throwing overboard the dead and dying , typhoon coming on ) , joseph mallord william turner ( video , images , additional resources ) 112 . palace of westminster ( houses of parliament ) , charles barry , a.w.n . pugin ( video , images , additional resources ) 113 . the stonebreakers , gustave courbet ( essay , image , additional resources 1 , 2 ) 114 . nadar elevating photography to art , honoré daumier ( essay , image , additionalresources ) 115 . olympia , édouard manet ( video , image , additional resources ) 116 . the saint-lazare station , claude monet ( video , image , additional resources 117 . the horse in motion , eadweard muybridge ( essay , image , additional resources ) 118 . the valley of mexico from the hillside of santa isabel , josé maría velasco ( video , essay , photos ) 119 . the burghers of calais , auguste rodin ( essay , image , 3d image , additional resources ) 120 . the starry night , vincent van gogh ( essay , image , additional resources ) 121 . the coiffure , mary cassatt ( essay , image , additional resources ) 122 . the scream , edvard munch ( essay , image , additional resources ) 123 . where do we come from ? what are we ? where are we going ? , paul gauguin ( essay , images , additional resources ) 124 . carson , pirie , scott and company building , louis sullivan ( essay , images , additionalresources ) 125 . mont sainte-victoire , paul cézanne ( video , essay , image , additional resources 126 . les demoiselles d'avignon , pablo picasso ( video , images , additional resources 127 . the steerage , alfred stieglitz ( essay , image , additional resources ) 128 . the kiss , gustav klimt ( video , image , additional resources ) 129 . the kiss , constantin brancusi ( video , images , additional resources ) 130 . the portuguese , georges braque ( essay , images , additional resources ) 131 . the goldfish , henri matisse ( essay , image , additional resources ) 132 . improvisation 28 ( second version ) , vasily kandinsky ( video , photo , additional resources 133 . self-portrait as a soldier , ernst ludwig kirchner ( essay , image , additional resources 134 . memorial sheet of karl liebknecht , käthe kollwitz ( essay , image , additional resources ) 135 . villa savoye , le corbusier ( essay , images , additional resources ) 136 . composition with red , blue and yellow , piet mondrian ( essay , image , additional resources ) 137 . illustration from the results of the first five-year plan , varvarastepanova ( essay , image , additional resources ) 138 . object ( le déjeuner en fourrure ) , meret oppenheim ( essay , quiz , image , additional resources 139 . fallingwater , frank lloyd wright ( essay , plans and elevations , images , additionalresources ) 140 . the two fridas , frida kahlo ( essay , image , additional resources ) 141 . the migration of the negro , panel no . 49 , jacob lawrence ( video-short version , video-long version , photo , additional resources ) 142 . the jungle , wilfredo lam ( essay , image , additional resources ) 143 . dream of a sunday afternoon in alameda central park , diego rivera ( essay , image , additional resources ) 144 . fountain , marcel duchamp ( video , images , additional resources ) 145 . woman i , willem de kooning ( video , images , additional resources ) 146 . seagram building , ludwig mies van der rohe , philip johnson ( video , images , additional resources ) 147 . marilyn diptych , andy warhol ( essay , image , additional resources ) 148 . narcissus garden , yayoi kusama ( essay , image , additional resources ) 149 . the bay , helen frankenthaler ( essay , quiz , image , additional resources ) 150 . lipstick ( ascending ) on caterpillar tracks , claes oldenburg ( essay , photo , additional resoures ) 151 . spiral jetty , robert smithson ( video , images , additional resources ) 152 . house in new castle county , robert ventura , john rausch and denise scott brown ( essay , photos , additional resources ) content area 5 indigenous americas 1000 b.c.e.-1980 c.e . 153 . chavín de huántar ( essay , image , additional resources ) 154 . mesa verde cliff dwellings ( essay , video , photos , additional resources ) 155 . yaxchilán lintel 25 , structure 23 ( essay , essay on related lintel , photo , additional resources ) 156 . great serpent mound ( essay , photos , additional resources ) 157 . templo mayor , main aztec temple ( video , essay , images , additional resources ) a . the coyolxauhqui stone ( video , images , additional resources ) b . calendar stone ( video , images , additional resources ) c. olmec-style mask ( video , images , additional resources ) 158 . ruler 's feather headdress ( probably of moctezuma ii ) ( video , images , additionalresources , see page 12 ) 159 . city of cusco ( essay , video , photos , additional resources ) 160 . maize cobs ( essay , image , additional resources 1 , 2 ) 161 . city of machu picchu ( essay , video , photos , additional resources ) 162 . all-t'oqapu tunic ( essay , photo , additional resources ) 163 . bandolier bag ( essay , photo , additional resources ) 164 . transformation mask ( essay , image , additional resources ) 165 . painted elk hide , attributed to cotsiogo ( cadzi cody ) ( essay , image , additionalresources ) 166 . black-on-black ceramic vessel , maria martínez and julian martínez ( essay , image , additional resources ) content area 6 africa 1100-1980 c.e . 167 . conical tower and circular wall of great zimbabwe ( video , images , additional resources ) 168 . great mosque of djenné ( essay , video , images , additional resources ) 169 . wall plaque , from oba 's palace ( essay , images , additional resources ) 170 . sika dwa kofi ( golden stool ) ( video , image , additional resources ) 171 . ndop ( portrait figure ) of king mishe mishyaang mambul ( essay , image , additional resources ) 172 . nkisi n ’ kondi ( essay , image , additional resources ) 173 . female ( pwo ) mask ( video , images , additional resources ) 174 . portrait mask ( mblo ) ( essay , image , additional resources ) 175 . bundu mask ( video , image , additional resources ) 176 . ikenga ( shrine figure ) ( video , image , additional resources ) 177 . lukasa ( memory board ) ( essay , image , additional resources ) 178 . aka elephant mask ( video , photos , additional resources 179 . reliquary figure ( byeri ) ( video , photos , additional resources ) 180 . veranda post of enthroned king and senior wife ( opo ogoga ) ( related video , image 1 , image 2 , additional resources ) content area 7 west and central asia 500 b.c.e.-1980 c.e . 181 . petra , jordan : treasury and great temple ( additional resources ) a. nabataeans introduction ( essay ) b. petra and the treasury ( essay , images ) c. petra and the great temple ( essay , images ) d. unesco siq project ( video ) 182 . buddha , bamiyan ( essay , image , video , additional resources 1 , 2 ) 183 . the kaaba ( essay , images , additional resources ) 184 . jowo rinpoche , enshrined in the jokhang temple ( essay , images , additional resources ) 185 . dome of the rock ( essay , images , additional resources ) 186 . great mosque ( masjid-e jameh ) , isfahan ( essay , images , additional resources ) 187 . folio from a qur'an ( essay , image , additional resources ) 188 . basin ( baptistère de saint louis ) , mohammed ibn al-zain ( video , images , additional resources ) 189 . bahram gur fights the karg , folio from the great il-khanid shahnama ( essay , image , additional resources ) 190 . the court of gayumars , folio from shah tahmasp's shahnama ( essay , image , additional resources ) 191 . the ardabil carpet ( essay , images , additional resources ) content area 8 south , east and southeast asia 300 b.c.e.-1980 c.e . 192 . great stupa at sanchi ( essay , video , images , additional resources ) 193 . terracotta warriors from mausoleum of the first qin emperor of china ( essay , video , images , additional resources ) 194 . funeral banner of lady dai ( xin zhui ) ( essay , images , additional resources 195 . longmen caves ( essay , video , images , additional resources ) 196 . gold and jade crown ( essay , image , additional resources ) 197 . todai-ji ( essay , video , images , additional resources ) 198 . borobudur ( essay , images , additional resources ) 199 . angkor , the temple of angkor wat , the city of angkor thom , cambodia ( essay , video , images , additional resources ) 200 . lakshmana temple ( essay , additional resources ) 201 . travelers among mountains and streams , fan kuan ( essay , image , additional resources ) 202 . shiva as lord of dance ( nataraja ) ( essay , image , additional resources ) 203 . night attack on the sanjô palace ( essay , photos , additional resources ) 204 . the david vases ( video , essay , images , additional resources 1 , 2 ) 205 . portrait of sin sukju ( essay , image , additional resources ) 206 . forbidden city ( essay , video , images , additional resources ) 207 . ryoan-ji ( video , essay , images , additional resources 1 , 2 ) 208 . jahangir preferring a sufi shaikh to kings , bichitr ( essay , image , additional resources ) 209 . taj mahal ( essay , images , additional resources ) 210 . white and red plum blossoms , ogata korin ( essay , image , additional resources 1 , additional resources 2 , additional resources 3 ) 211 . under the wave off kanagawa ( kanagawa oki nami ura ) , also known as the great wave , from the series thirty-six views of mount fuji , katsushika hokusai ( essay , image , additional resources ) 212 . chairman mao en route to anyuan ( essay , image additional resources ) content area 9 the pacific 700-1980 c.e . 213 . nan madol 214 . moai on platform ( ahu ) ( video , essay , images , additional resources 1 , 2 , 3 ) 215 . 'ahu 'ula ( feather cape ) ( essay , image , additional resources ) 216 . staff god ( essay , images , additional resources ) 217 . female deity from nukuoro ( essay , images , additional resources 1 , 2 , 3 ) 218 . buk mask ( video , images , additional resources ) 219 . hiapo ( tapa ) ( essay , image , additional resources ) 220 . tamati waka nene , gottfried lindaur ( essay , image , additional resources 221 . navigation chart ( essay , image , additional resources ) 222 . malagan display and mask ( essay , image 1 , 2 , additional resources ) 223 . presentation of fijian mats and tapas cloths to queen elizabeth ii ( essay , image , additional resources 1 , 2 ) content area 10 global contemporary 1980 c.e . to present 224 . the gates , christo and jeanne-claude ( essay , images , additional resources ) 225 . vietnam veterans memorial , maya lin ( video , images , additional resources ) 226 . horn players , jean-michel basquiat ( essay , image , additional resources ) 227 . summer trees , song su-nam ( essay , image , additional resources ) 228 . androgyne iii , magdalena abakanowicz ( essay , images , additional resources ) 229 . a book from the sky , xu bing ( video , images , additional resources 230 . pink panther , jeff koons ( essay , images , additional resources ) 231 . untitled ( # 228 ) , from the history portraits series , cindy sherman ( essay , image , additional resources ) 232 . dancing at the louvre , from the series , the french collection , part 1 ; # 1 , faith ringgold ( essay , image , additional resources ) 233 . trade ( gifts for trading land with white people ) , jaune quick-to-see smith ( essay , image , additional resources ) 234 . earth ’ s creation , emily kame kngwarreye ( essay , image , additional resources ) 235 . rebellious silence , from the women of allah series , shirin neshat ( artist ) ; photo by cynthia preston ( essay , image , additional resources ) 236 . en la barberia no se llora ( no crying allowed in the barbershop ) , pepon osorio ( essay , image 1 , 2 , additional resources ) 237 . pisupo lua afe ( corned beef 2000 ) , michel tuffery ( essay , artist , biography , video 1 , video 2 , images ) 238 . electronic superhighway , nam june paik ( essay , image , additional resources 239 . the crossing , bill viola ( essay , image , additional resources ) 240 . guggenheim museum bilbao , frank gehry ( essay , images , additional resources ) 241 . pure land , mariko mori ( essay , image , additional resources ) 242 . lying with the wolf , kiki smith ( essay , image , additional resources ) 243.darkytown rebellion , kara walker ( essay , image , additional resources ) 244 . the swing ( after fragonard ) , yinka shonibare ( essay , images , additional resources ) 245 . old man ’ s cloth , el anatsui ( video , essay , image , additional resources ) 246 . stadia ii , julie mehretu ( essay , image , additional resources ) 247 . preying mantra , wangechi mutu ( essay , image , additional resources ) 248 . shibboleth , doris salcedo ( essay , images , additional resources ) 249 . maxxi national museum of xxi century arts , zaha hadid ( video , images , additional resources ) 250 . kui hua zi ( sunflower seeds ) , ai weiwei ( essay , video , images , additional resources ) special thanks to the many art historians and curators who have contributed their expertise as well as our museum partners , the american museum of natural history , the asian art museum , the british museum , the j. paul getty museum , the metropolitan museum of art , the museum of modern art , and tate . ap art history is a registered trademark of the college board , which was not involved in the production of , and does not endorse , this product .
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smarthistory is adding new materials regularly . for more context on these works of art , please see the art history sections of khan academy . content area 1 global prehistory 30,000-500b.c.e .
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are there any works of art from the philippines ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop .
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in the `` kirchhoff 's voltage law - concept check '' , why is the `` -1v '' negative ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow .
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when i see the words `` voltage on each resistor '' - what does that mean ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow .
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that voltage is getting eaten up or used by the resistor ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero .
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what is the algebraic sum ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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$ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you .
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what if all the current ( arrows ) are pointing inward , how is the sum of current zero in that case ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages .
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what if the resistors are in parallel ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways .
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is kirchhoff 's law applicable for ac circuits ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ .
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i mean when you see a diagram its usually done clock wise either way , correct ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . )
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and does in play a role in adding up of the voltages.. ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop .
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how do we apply kirchhoff 's laws to parallel circuits ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element .
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how do you label the signs around each resistor based on what ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations .
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if we calculate the sum of voltages with the answer 6v , we will get 5+3+6+ ( -1 ) = 13 and the source have 15 v. where are the two missing volts ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ?
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hello , normal electrical engineering problems are not labeled with positive and negative signs around the resistors , how would you solve problems with no labeling of positive and negative signs using kvl and kcl ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements .
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why is there voltage difference across the resistors ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ?
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is resistance is ever positive ... ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ?
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while making up these problems , how do you decide which side of the resistor is the negative terminal and which one is the positive terminal ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go .
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could you also explain how this will apply to loops with parallel circuits ?
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kirchhoff 's laws for current and voltage lie at the heart of circuit analysis . with these two laws , plus the equations for individual component ( resistor , capacitor , inductor ) , we have the basic tool set we need to start analyzing circuits . this article assumes you are familiar with the definitions of node , distributed node , branch , and loop . you may want to have a pencil and paper nearby to work the example problems . currents into a node try to reason through this example by yourself , before we talk about the theory . the schematic below shows four branch currents flowing in and out of a distributed node . the various currents are in milliamps , $ \text { ma } $ . one of the currents , $ \blued i $ , is not known . problem 1 : what is $ i $ ? here 's another example , this time with variable names instead of numerical values . this node happens to have $ 5 $ branches . each branch might ( or might not ) carry a current , labeled $ i_1 \ , \text { to } \ , i_5 $ . all the arrows are drawn pointing in . this choice of direction is arbitrary . arrows pointing inward is as good a choice as any at this point . the arrows establish a reference direction for what we choose to call a positive current . look at branch current $ { i_1 } $ . where does it go ? the first thing $ { i_1 } $ does is flow into the node ( represented by the black dot ) . then what ? here 's two things $ { i_1 } $ ca n't do : the flowing charge in $ { i_1 } $ ca n't stay inside the node . ( the node does not have a place to store charge ) . and $ { i_1 } $ 's charge ca n't jump off the wires into thin air . charge just does n't do that under normal circumstances . what 's left ? : the current has to flow out of the node through one or more of the other branches . for our example node , we would write this as , $ i_1 + i_2 + i_3 + i_4 + i_5 = 0 $ if $ { i_1 } $ is a positive current flowing into the node , then one or more of the other currents must be flowing out . those outgoing currents will have a $ - $ negative sign . this observation about currents flowing in a node is nicely captured in general form as kirchhoff 's current law . kirchhoff 's current law the sum of all currents flowing into a node is zero . kirchhoff 's current law can be written as , $ \large\displaystyle \sum_n i_n = 0 $ the index $ n $ counts the branches attached to the node . kirchhoff 's current law is flexible . it can be stated other ways . we could phrase it as : the sum of all currents flowing out of a node is zero . the summation equation for this form of the law is the same as above . just draw all the current arrows pointing out of the node . it is also correct to say : the sum of all currents flowing into a node equals the sum of currents flowing out of the node . $ \large \displaystyle \sum i_ { in } = \sum i_ { out } $ this form of the law requires you to do a little more bookkeeping on the current arrows . current arrows all in ? if you are wondering how all currents can add up to zero if they all point in the same direction , do not let this bother you . arrows are reference directions . when the circuit analysis is complete , the math will make sure one ( or more ) of the branch currents has a negative sign , and flows in the opposite direction of its arrow . kirchhoff 's current law - concept checks currents are in milliamps , $ \text { ma } $ . problem 2 : what is $ i_5 $ ? problem 3 : what is $ i_3 $ in this distributed node ? voltage around a loop below is a circuit with four resistors and a voltage source . we will solve this from scratch using ohm 's law . then we will look at the result and make some observations . the first step in solving the circuit is to compute the current . then we will compute the voltage across the individual resistors . we recognize this as a series circuit , so there is only one current flowing , $ \blued i $ , through all five elements . to find $ \blued i $ , the four series resistors can be reduced to a single equivalent resistor : $ r_ { series } = 100 + 200 + 300 + 400 = 1000\ , \omega $ using ohm 's law , the current is : $ \blued i = \dfrac { v } { r_ { series } } = \dfrac { 20\ , \text v } { 1000\ , \omega } = 0.020\ , \text a = 20 \ , \text { ma } $ now we know the current . next we find the voltages across the four resistors . go back to the original schematic and add voltage labels to all five elements : apply ohm 's law four more times to find the voltage across each resistor : $ v\phantom { { \text { r1 } } } = \blued i\ , \text r $ $ v { \text { r1 } } = 20\ , \text { ma } \cdot 100\ , \omega = +2\ , \text { v } $ $ v_ { \text { r2 } } = 20\ , \text { ma } \cdot 200\ , \omega = +4\ , \text { v } $ $ v_ { \text { r3 } } = 20\ , \text { ma } \cdot 300\ , \omega = +6\ , \text { v } $ $ v_ { \text { r4 } } = 20\ , \text { ma } \cdot 400\ , \omega = +8\ , \text { v } $ we know the current and all voltages . the circuit is now solved . we can write the voltages for the resistors and the source on the schematic . these five voltages are referred to as element voltages . ( the circuit nodes get names , $ \greene { \text a } $ to $ \greene { \text e } $ , so we can talk about them . ) let 's do a quick check . add up the voltages across the resistors , $ 2\ , \text { v } + 4\ , \text { v } + 6\ , \text { v } + 8\ , \text { v } = 20 \ , \text v $ the individual resistor voltages add up to the source voltage . this makes sense , and confirms our calculations . now we add up the voltages again , using a slightly different procedure : by `` going around the loop . '' there 's no new science here , we are just rearranging the same computation . procedure : add element voltages around a loop step 1 : pick a starting node . step 2 : pick a direction to travel around the loop ( clockwise or counterclockwise ) . step 3 : walk around the loop . include element voltages in a growing sum according to these rules : when you encounter a new element , look at the voltage sign as you enter the element . if the sign is $ + $ , then there will be a voltage drop going through the element . subtract the element voltage . if the sign is $ - $ , then there will be a voltage rise going through the element . add the element voltage . step 4 : continue around the loop until you reach the starting point , including element voltages all the way around . apply the loop procedure let 's follow the procedure step-by-step . start at the lower left at node $ \greene { \text a } $ . walk clockwise . the first element we come to is the voltage source . the first voltage sign we encounter is a $ - $ minus sign , so there is going to be a voltage rise going through this element . consulting the procedure step 3. , we initialize the loop sum by adding the source voltage . $ $ $ v_ { loop } = +20\ , \text v $ going through the voltage source , to node $ \greene { \text b } $ . the next element we encounter is the $ 100\ , \omega $ resistor . its nearest voltage sign is $ + $ . consult the procedure again , and this time we subtract the element voltage from the growing sum . $ v_ { loop } = + 20\ , \text v - 2\ , \text v $ going through the $ 100\ , \omega $ resistor , to node $ \greene { \text c } $ . keep going . next we visit the $ 200\ , \omega $ resistor , and again we first encounter a $ + $ sign , so we subtract this voltage . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v $ going through the $ 200\ , \omega $ resistor , to node $ \greene { \text d } $ . we complete the loop with the addition of two more elements , $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v \ , $ through the $ 300\ , \omega $ resistor , to node $ \greene { \text e } $ . $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v\ , $ after the $ 400\ , \omega $ resistor . ( check the circuit diagram , make sure i got the last two $ - $ signs correct . ) done . we made it back home to node $ \greene { \text a } $ . what does this expression for $ v_ { loop } $ add up to ? $ $ $ v_ { loop } = + 20\ , \text v - 2\ , \text v - 4\ , \text v - 6\ , \text v - 8\ , \text v = 0 $ the sum of voltages going around the loop is $ 0 $ . the starting and ending node is the same , so the starting and ending voltage is the same . on your `` walk '' you went up voltage rises and down voltage drops , and they all cancel out when you get back to where you started . this happens because electric force is conservative . there is n't a net gain or loss of energy if you return to the same place you started . we 'll do another example , this time with variable names instead of numerical values . the following familiar schematic is labeled with voltages and node names . the voltage polarity on the resistors is arranged in a way you might not expect , with all the arrows pointing in the same direction around the loop . this reveals a cool property of loops . let 's take a walk around the loop , adding up voltages as we go . our starting point is node $ \greene { \text a } $ in the lower left corner . our walk goes clockwise around the loop ( an arbitrary choice , either way works ) . starting at node $ \greene { \text a } $ , going up , we first encounter a minus sign on the voltage source , which says there is going to be a voltage rise of $ v_ { ab } $ volts going through the voltage source . because it is a voltage rise , this element voltage gets a $ + $ sign when we include it in the loop sum . continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign . the final loop sum looks like this : $ +v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } $ what does this add up to ? let 's reason it out . the loop starts and ends at the same node , so the starting and ending voltages are identical . we went around the loop , adding voltages , and we end up back at the same voltage . that means the voltages have to add to zero . for our example loop , we would write this as , $ v_ { \text { ab } } + v_ { \text { r1 } } + v_ { \text { r2 } } + v_ { \text { r3 } } + v_ { \text { r4 } } = 0 $ this observation about voltages around a loop is nicely captured in general form as kirchhoff 's voltage law . kirchhoff 's voltage law kirchhoff 's voltage law : the sum of voltages around a loop is zero . kirchhoff 's voltage law can be written as , $ \large\displaystyle \sum_n v_n = 0 $ where $ n $ counts the element voltages around the loop . you can also state kirchhoff 's voltage law another way : the sum of voltage rises equals the sum of voltage drops around a loop . $ \large \displaystyle \sum v_ { rise } = \sum v_ { drop } $ kirchhoff 's voltage law has some nice properties : you can trace a loop starting from any node . walk around the loop and end up back at the starting node , the sum of voltages around the loop adds up to zero . you can go around the loop in either direction , clockwise or counterclockwise . kirchhoff 's voltage law still holds . if a circuit has multiple loops , kirchhoff 's voltage law is true for every loop . voltages all positive ? if you are wondering : how can the element voltages all be positive if they have to add up to zero ? it 's okay . the voltage arrows and polarity signs are just reference directions for voltage . when the circuit analysis is complete , one or more of the element voltages around the loop will be negative with respect to its voltage arrow . the signs of the actual voltages always sort themselves out during calculations . kirchhoff 's voltage law - concept check problem 4 : what is $ v_ { r3 } $ ? reminder : check the first sign of each element voltage as you walk around the loop . summary we were introduced to two new friends . kirchhoff 's current law for branch currents at a node , $ \large\displaystyle \sum_n i_n = 0 $ kirchhoff 's voltage law for element voltages around a loop , $ \large\displaystyle \sum_n v_n = 0 $ our new friends sometimes go by their initials , kcl and kvl . and we learned it 's important to pay close attention to voltage and current signs if we want correct answers . this is a tedious process that requires attention to detail . it is a core skill of a good electrical engineer .
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continue around the loop from node $ \greene { \text b } $ to $ \greene { \text c } $ to $ \greene { \text d } $ to $ \greene { \text e } $ , and finish back home at node $ \greene { \text a } $ . append resistor voltages to the loop sum along the way . the polarity labels on all the resistors are arranged so we encounter a $ - $ sign as we approach each resistor . so the resistor voltages all go into the loop sum with a $ + $ sign .
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if we just connect battery with the couple of resistors will the resistor dissipate the energy or in other words will our battery end up after some time ?
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key points : primary producers ( usually plants and other photosynthesizers ) are the gateway for energy to enter food webs . productivity is the rate at which energy is added to the bodies of a group of organisms ( such as primary producers ) in the form of biomass . gross productivity is the overall rate of energy capture . net productivity is lower , adjusted for energy used by organisms in respiration/metabolism . energy transfer between trophic levels is inefficient . only $ \sim10\ % $ of the net productivity of one level ends up as net productivity at the next level . ecological pyramids are visual representations of energy flow , biomass accumulation , and number of individuals at different trophic levels . introduction have you ever wondered what would happen if all the plants on earth disappeared ( along with other photosynthesizers , like algae and bacteria ) ? well , our beautiful planet would definitely look barren and sad . we would also lose our main source of oxygen ( that important stuff we breathe and rely on for metabolism ) . carbon dioxide would no longer be cleaned out of the air , and as it trapped heat , earth might warm up fast . and , perhaps most problematically , almost every living thing on earth would eventually run out of food and die . why would this be the case ? in almost all ecosystems , photosynthesizers are the only `` gateway '' for energy to flow into food webs ( networks of organisms that eat one another ) . if photosynthesizers were removed , the flow of energy would be cut off , and the other organisms would run out of food . in this way , photosynthesizers lay the foundation for every light-receiving ecosystem . producers are the energy gateway plants , algae , and photosynthetic bacteria act as producers . producers are autotrophs , or `` self-feeding '' organisms , that make their own organic molecules from carbon dioxide . photoautotrophs like plants use light energy to build sugars out of carbon dioxide . the energy is stored in the chemical bonds of the molecules , which are used as fuel and building material by the plant . the energy stored in organic molecules can be passed to other organisms in the ecosystem when those organisms eat plants ( or eat other organisms that have previously eaten plants ) . in this way , all the consumers , or heterotrophs ( `` other-feeding '' organisms ) of an ecosystem , including herbivores , carnivores , and decomposers , rely on the ecosystem 's producers for energy . if the plants or other producers of an ecosystem were removed , there would be no way for energy to enter the food web , and the ecological community would collapse . that 's because energy is n't recycled : instead , it 's dissipated as heat as it moves through the ecosystem , and must be constantly replenished . because producers support all the other organisms in an ecosystem , producer abundance , biomass ( dry weight ) , and rate of energy capture are key in understanding how energy moves through an ecosystem and what types and numbers of other organisms it can sustain . primary productivity in ecology , productivity is the rate at which energy is added to the bodies of organisms in the form of biomass . biomass is simply the amount of matter that 's stored in the bodies of a group of organisms . productivity can be defined for any trophic level or other group , and it may take units of either energy or biomass . there are two basic types of productivity : gross and net . to illustrate the difference , let 's consider primary productivity ( the productivity of the primary producers of an ecosystem ) . gross primary productivity , or gpp , is the rate at which solar energy is captured in sugar molecules during photosynthesis ( energy captured per unit area per unit time ) . producers such as plants use some of this energy for metabolism/cellular respiration and some for growth ( building tissues ) . net primary productivity , or npp , is gross primary productivity minus the rate of energy loss to metabolism and maintenance . in other words , it 's the rate at which energy is stored as biomass by plants or other primary producers and made available to the consumers in the ecosystem . plants typically capture and convert about $ 1.3 $ $ \mbox { - } $ $ 1.6\ % $ of the solar energy that reaches earth 's surface and use about a quarter of the captured energy for metabolism and maintenance . so , around $ 1\ % $ of the solar energy reaching earth 's surface ( per unit area and time ) ends up as net primary productivity . net primary productivity varies among ecosystems and depends on many factors . these include solar energy input , temperature and moisture levels , carbon dioxide levels , nutrient availability , and community interactions ( e.g. , grazing by herbivores ) $ ^2 $ . these factors affect how many photosynthesizers are present to capture light energy and how efficiently they can perform their role . in terrestrial ecosystems , primary productivity ranges from about $ 2 , $ $ 000 $ $ \text { g/m } ^2\text { /yr } $ in highly productive tropical forests and salt marshes to less than $ 100 $ $ \text { g/m } ^2\text { /yr } $ in some deserts . you can see how net primary productivity changes on shorter timescales in the dynamic map below , which shows seasonal and year-to-year variations in net primary productivity of terrestrial ecosystems across the globe . how does energy move between trophic levels ? energy can pass from one trophic level to the next when organic molecules from an organism 's body are eaten by another organism . however , the transfer of energy between trophic levels is not usually very efficient . how inefficient ? on average , only about $ 10\ % $ of the energy stored as biomass in one trophic level ( e.g. , primary producers ) gets stored as biomass in the next trophic level ( e.g. , primary consumers ) . put another way , net productivity usually drops by a factor of ten from one trophic level to the next . for example , in one aquatic ecosystem in silver springs , florida , the net productivities ( rates of energy storage as biomass ) for trophic levels were $ ^3 $ : primary producers , such as plants and algae : $ 7618 $ $ \text { kcal/m } ^2\text { /yr } $ primary consumers , such as snails and insect larvae : $ 1103 $ $ \text { kcal/m } ^2\text { /yr } $ secondary consumers , such as fish and large insects : $ 111 $ $ \text { kcal/m } ^2\text { /yr } $ tertiary consumers , such as large fish and snakes : $ 5 $ $ \text { kcal/m } ^2\text { /yr } $ transfer efficiency varies between levels and is not exactly $ 10\ % $ , but we can see that it 's in the ballpark by doing a few calculations . for instance , the efficiency of transfer between primary producers and primary consumers is : $ \text { transfer efficiency } = $ $ \dfrac { 1103\ : \ : \text { kcal/m } ^2\text { /yr } } { 7618\ : \ : \text { kcal/m } ^2\text { /yr } } \times 100 $ $ \text { transfer efficiency } = 14.5\ % $ why is energy transfer inefficient ? there are several reasons . one is that not all the organisms at a lower trophic level get eaten by those at a higher trophic level . another is that some molecules in the bodies of organisms that do get eaten are not digestible by predators and are lost in the predators ' feces ( poop ) . the dead organisms and feces become dinner for decomposers . finally , of the energy-carrying molecules that do get absorbed by predators , some are used in cellular respiration ( instead of being stored as biomass ) $ ^ { 4,5 } $ . want to put some concrete numbers behind these concepts ? click on the pop-up to see exactly where energy goes as it moves through the silver springs ecosystem : ecological pyramids we can look at numbers and do calculations to see how energy flows through an ecosystem . but would n't it be nice to have a diagram that captures this information in an easy-to-process way ? ecological pyramids provide an intuitive , visual picture of how the trophic levels in an ecosystem compare for a feature of interest ( such as energy flow , biomass , or number of organisms ) . let 's take a look at these three types of pyramids and see how they reflect the structure and function of ecosystems . energy pyramids energy pyramids represent energy flow through trophic levels . for instance , the pyramid below shows gross productivity for each trophic level in the silver springs ecosystem . an energy pyramid usually shows rates of energy flow through trophic levels , not absolute amounts of energy stored . it can have energy units , such as $ \text { kcal/m } ^2\text { /yr } $ , or biomass units , such as $ \text { g/m } ^2\text { /yr } $ . energy pyramids are always upright , that is , narrower at each successive level ( unless organisms enter the ecosystem from elsewhere ) . this pattern reflects the laws of thermodynamics , which tell us that new energy ca n't be created , and that some must be converted to a not-useful form ( heat ) in each transfer . biomass pyramids another way to visualize ecosystem structure is with biomass pyramids . these pyramids represent the amount of energy that 's stored in living tissue at the different trophic levels . ( unlike energy pyramids , biomass pyramids show how much biomass is present in a level , not the rate at which it 's added . ) below on the left , we can see a biomass pyramid for the silver springs ecosystem . this pyramid , like many biomass pyramids , is upright . however , the biomass pyramid shown on the right – from a marine ecosystem in the english channel – is upside-down ( inverted ) . the inverted pyramid is possible because of the high turnover rate of the phytoplankton . they get rapidly eaten by the primary consumers ( zooplankton ) , so their biomass at any point in time is small . however , they reproduce so fast that , despite their low steady-state biomass , they have high primary productivity that can support large numbers of zooplankton . numbers pyramids numbers pyramids show how many individual organisms there are in each trophic level . they can be upright , inverted , or kind of lumpy , depending on the ecosystem . as shown in the figure below , a typical grassland during the summer has a base of numerous plants , and the numbers of organisms decrease at higher trophic levels . however , during the summer in a temperate forest , the base of the pyramid instead consists of a few plants ( mostly trees ) that are vastly outnumbered by primary consumers ( mostly insects ) . because individual trees are big , they can support the other trophic levels despite their small numbers . summary primary producers , which are usually plants and other photosynthesizers , are the gateway through which energy enters food webs . productivity is the rate at which energy is added to the bodies of a group of organisms , such as primary producers , in the form of biomass . gross productivity is the overall rate of energy capture . net productivity is lower : it 's gross productivity adjusted for the energy used by the organisms in respiration/metabolism , so it reflects the amount of energy stored as biomass . energy transfer between trophic levels is not very efficient . only $ \sim10\ % $ of the net productivity of one level ends up as net productivity at the next level . ecological pyramids are visual representations of energy flow , biomass accumulation , and number of individuals at different trophic levels .
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in this way , all the consumers , or heterotrophs ( `` other-feeding '' organisms ) of an ecosystem , including herbivores , carnivores , and decomposers , rely on the ecosystem 's producers for energy . if the plants or other producers of an ecosystem were removed , there would be no way for energy to enter the food web , and the ecological community would collapse . that 's because energy is n't recycled : instead , it 's dissipated as heat as it moves through the ecosystem , and must be constantly replenished .
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my name is jannah , my question is are the decomposer 's the main part of the food chain/ the food web ?
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key points : primary producers ( usually plants and other photosynthesizers ) are the gateway for energy to enter food webs . productivity is the rate at which energy is added to the bodies of a group of organisms ( such as primary producers ) in the form of biomass . gross productivity is the overall rate of energy capture . net productivity is lower , adjusted for energy used by organisms in respiration/metabolism . energy transfer between trophic levels is inefficient . only $ \sim10\ % $ of the net productivity of one level ends up as net productivity at the next level . ecological pyramids are visual representations of energy flow , biomass accumulation , and number of individuals at different trophic levels . introduction have you ever wondered what would happen if all the plants on earth disappeared ( along with other photosynthesizers , like algae and bacteria ) ? well , our beautiful planet would definitely look barren and sad . we would also lose our main source of oxygen ( that important stuff we breathe and rely on for metabolism ) . carbon dioxide would no longer be cleaned out of the air , and as it trapped heat , earth might warm up fast . and , perhaps most problematically , almost every living thing on earth would eventually run out of food and die . why would this be the case ? in almost all ecosystems , photosynthesizers are the only `` gateway '' for energy to flow into food webs ( networks of organisms that eat one another ) . if photosynthesizers were removed , the flow of energy would be cut off , and the other organisms would run out of food . in this way , photosynthesizers lay the foundation for every light-receiving ecosystem . producers are the energy gateway plants , algae , and photosynthetic bacteria act as producers . producers are autotrophs , or `` self-feeding '' organisms , that make their own organic molecules from carbon dioxide . photoautotrophs like plants use light energy to build sugars out of carbon dioxide . the energy is stored in the chemical bonds of the molecules , which are used as fuel and building material by the plant . the energy stored in organic molecules can be passed to other organisms in the ecosystem when those organisms eat plants ( or eat other organisms that have previously eaten plants ) . in this way , all the consumers , or heterotrophs ( `` other-feeding '' organisms ) of an ecosystem , including herbivores , carnivores , and decomposers , rely on the ecosystem 's producers for energy . if the plants or other producers of an ecosystem were removed , there would be no way for energy to enter the food web , and the ecological community would collapse . that 's because energy is n't recycled : instead , it 's dissipated as heat as it moves through the ecosystem , and must be constantly replenished . because producers support all the other organisms in an ecosystem , producer abundance , biomass ( dry weight ) , and rate of energy capture are key in understanding how energy moves through an ecosystem and what types and numbers of other organisms it can sustain . primary productivity in ecology , productivity is the rate at which energy is added to the bodies of organisms in the form of biomass . biomass is simply the amount of matter that 's stored in the bodies of a group of organisms . productivity can be defined for any trophic level or other group , and it may take units of either energy or biomass . there are two basic types of productivity : gross and net . to illustrate the difference , let 's consider primary productivity ( the productivity of the primary producers of an ecosystem ) . gross primary productivity , or gpp , is the rate at which solar energy is captured in sugar molecules during photosynthesis ( energy captured per unit area per unit time ) . producers such as plants use some of this energy for metabolism/cellular respiration and some for growth ( building tissues ) . net primary productivity , or npp , is gross primary productivity minus the rate of energy loss to metabolism and maintenance . in other words , it 's the rate at which energy is stored as biomass by plants or other primary producers and made available to the consumers in the ecosystem . plants typically capture and convert about $ 1.3 $ $ \mbox { - } $ $ 1.6\ % $ of the solar energy that reaches earth 's surface and use about a quarter of the captured energy for metabolism and maintenance . so , around $ 1\ % $ of the solar energy reaching earth 's surface ( per unit area and time ) ends up as net primary productivity . net primary productivity varies among ecosystems and depends on many factors . these include solar energy input , temperature and moisture levels , carbon dioxide levels , nutrient availability , and community interactions ( e.g. , grazing by herbivores ) $ ^2 $ . these factors affect how many photosynthesizers are present to capture light energy and how efficiently they can perform their role . in terrestrial ecosystems , primary productivity ranges from about $ 2 , $ $ 000 $ $ \text { g/m } ^2\text { /yr } $ in highly productive tropical forests and salt marshes to less than $ 100 $ $ \text { g/m } ^2\text { /yr } $ in some deserts . you can see how net primary productivity changes on shorter timescales in the dynamic map below , which shows seasonal and year-to-year variations in net primary productivity of terrestrial ecosystems across the globe . how does energy move between trophic levels ? energy can pass from one trophic level to the next when organic molecules from an organism 's body are eaten by another organism . however , the transfer of energy between trophic levels is not usually very efficient . how inefficient ? on average , only about $ 10\ % $ of the energy stored as biomass in one trophic level ( e.g. , primary producers ) gets stored as biomass in the next trophic level ( e.g. , primary consumers ) . put another way , net productivity usually drops by a factor of ten from one trophic level to the next . for example , in one aquatic ecosystem in silver springs , florida , the net productivities ( rates of energy storage as biomass ) for trophic levels were $ ^3 $ : primary producers , such as plants and algae : $ 7618 $ $ \text { kcal/m } ^2\text { /yr } $ primary consumers , such as snails and insect larvae : $ 1103 $ $ \text { kcal/m } ^2\text { /yr } $ secondary consumers , such as fish and large insects : $ 111 $ $ \text { kcal/m } ^2\text { /yr } $ tertiary consumers , such as large fish and snakes : $ 5 $ $ \text { kcal/m } ^2\text { /yr } $ transfer efficiency varies between levels and is not exactly $ 10\ % $ , but we can see that it 's in the ballpark by doing a few calculations . for instance , the efficiency of transfer between primary producers and primary consumers is : $ \text { transfer efficiency } = $ $ \dfrac { 1103\ : \ : \text { kcal/m } ^2\text { /yr } } { 7618\ : \ : \text { kcal/m } ^2\text { /yr } } \times 100 $ $ \text { transfer efficiency } = 14.5\ % $ why is energy transfer inefficient ? there are several reasons . one is that not all the organisms at a lower trophic level get eaten by those at a higher trophic level . another is that some molecules in the bodies of organisms that do get eaten are not digestible by predators and are lost in the predators ' feces ( poop ) . the dead organisms and feces become dinner for decomposers . finally , of the energy-carrying molecules that do get absorbed by predators , some are used in cellular respiration ( instead of being stored as biomass ) $ ^ { 4,5 } $ . want to put some concrete numbers behind these concepts ? click on the pop-up to see exactly where energy goes as it moves through the silver springs ecosystem : ecological pyramids we can look at numbers and do calculations to see how energy flows through an ecosystem . but would n't it be nice to have a diagram that captures this information in an easy-to-process way ? ecological pyramids provide an intuitive , visual picture of how the trophic levels in an ecosystem compare for a feature of interest ( such as energy flow , biomass , or number of organisms ) . let 's take a look at these three types of pyramids and see how they reflect the structure and function of ecosystems . energy pyramids energy pyramids represent energy flow through trophic levels . for instance , the pyramid below shows gross productivity for each trophic level in the silver springs ecosystem . an energy pyramid usually shows rates of energy flow through trophic levels , not absolute amounts of energy stored . it can have energy units , such as $ \text { kcal/m } ^2\text { /yr } $ , or biomass units , such as $ \text { g/m } ^2\text { /yr } $ . energy pyramids are always upright , that is , narrower at each successive level ( unless organisms enter the ecosystem from elsewhere ) . this pattern reflects the laws of thermodynamics , which tell us that new energy ca n't be created , and that some must be converted to a not-useful form ( heat ) in each transfer . biomass pyramids another way to visualize ecosystem structure is with biomass pyramids . these pyramids represent the amount of energy that 's stored in living tissue at the different trophic levels . ( unlike energy pyramids , biomass pyramids show how much biomass is present in a level , not the rate at which it 's added . ) below on the left , we can see a biomass pyramid for the silver springs ecosystem . this pyramid , like many biomass pyramids , is upright . however , the biomass pyramid shown on the right – from a marine ecosystem in the english channel – is upside-down ( inverted ) . the inverted pyramid is possible because of the high turnover rate of the phytoplankton . they get rapidly eaten by the primary consumers ( zooplankton ) , so their biomass at any point in time is small . however , they reproduce so fast that , despite their low steady-state biomass , they have high primary productivity that can support large numbers of zooplankton . numbers pyramids numbers pyramids show how many individual organisms there are in each trophic level . they can be upright , inverted , or kind of lumpy , depending on the ecosystem . as shown in the figure below , a typical grassland during the summer has a base of numerous plants , and the numbers of organisms decrease at higher trophic levels . however , during the summer in a temperate forest , the base of the pyramid instead consists of a few plants ( mostly trees ) that are vastly outnumbered by primary consumers ( mostly insects ) . because individual trees are big , they can support the other trophic levels despite their small numbers . summary primary producers , which are usually plants and other photosynthesizers , are the gateway through which energy enters food webs . productivity is the rate at which energy is added to the bodies of a group of organisms , such as primary producers , in the form of biomass . gross productivity is the overall rate of energy capture . net productivity is lower : it 's gross productivity adjusted for the energy used by the organisms in respiration/metabolism , so it reflects the amount of energy stored as biomass . energy transfer between trophic levels is not very efficient . only $ \sim10\ % $ of the net productivity of one level ends up as net productivity at the next level . ecological pyramids are visual representations of energy flow , biomass accumulation , and number of individuals at different trophic levels .
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numbers pyramids numbers pyramids show how many individual organisms there are in each trophic level . they can be upright , inverted , or kind of lumpy , depending on the ecosystem . as shown in the figure below , a typical grassland during the summer has a base of numerous plants , and the numbers of organisms decrease at higher trophic levels .
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the pyramid of number for a marine ecosystem upright or inverted ?
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aesthetic refers to a type of human experience that combines perception , feeling , meaning making , and appreciation of qualities of produced and/or manipulated objects , acts , and events of daily life . aesthetic experience motivates behavior and creates categories through which our experiences of the world can be organized . artistic associations include self-defined groups , workshops , academies , and movements . artistic traditions are norms of artistic production and artistic products . artistic traditions are demonstrated through art-making processes ( utilization of materials and techniques , mode of display ) , through interactions between works of art and audience , and within form and/or content of a work of art . artistic changes are divergences from tradition in artistic choices demonstrated through art-making processes , through interactions between works of art and audience , and within form and/or content . tradition and change in form and content may be described in terms of style . audiences of a work of art are those who interact with the work as participants , facilitators , and/or observers . audience characteristics include gender , ethnicity , race , age , socioeconomic status , beliefs , and values . audience groups may be contemporaries , descendants , collectors , scholars , gallery/museum visitors , and other artists . content of a work of art consists of interacting , communicative elements of design , representation , and presentation within a work of art . content includes subject matter : visible imagery that may be formal depictions ( e.g. , minimalist or nonobjective works ) , representative depictions ( e.g. , portraiture and landscape ) , and/or symbolic depictions ( e.g. , emblems and logos ) . content may be narrative , symbolic , spiritual , historical , mythological , supernatural , and/or propagandistic ( e.g. , satirical and/or protest oriented ) . context includes original and subsequent historical and cultural milieu of a work of art . context includes information about the time , place , and culture in which a work of art was created , as well as information about when , where , and how subsequent audiences interacted with the work . the artist ’ s intended purpose for a work of art is contextual information , as is the chosen site for the work ( which may be public or private ) , as well as subsequent locations of the work . modes of display of a work of art can include associated paraphernalia ( e.g. , ceremonial objects and attire ) and multisensory stimuli ( e.g. , scent and sound ) . characteristics of the artist and audience—including aesthetic , intellectual , religious , political , social , and economic characteristics—are context . patronage , ownership of a work of art , and other power relationships are also aspects of context . contextual information includes audience response to a work of art . contextual information may be provided through records , reports , religious chronicles , personal reflections , manifestos , academic publications , mass media , sociological data , cultural studies , geographic data , artifacts , narrative and/or performance ( e.g. , oral , written , poetry , music , dance , dramatic productions ) , documentation , archaeology , and research . design elements are line , shape , color ( hue , value , saturation ) , texture , value ( shading ) , space , and form . design principles are balance/symmetry , rhythm/pattern , movement , harmony , contrast , emphasis , proportion/scale , and unity . form describes component materials and how they are employed to create physical and visual elements that coalesce into a work of art . form is investigated by applying design elements and principles to analyze the work ’ s fundamental visual components and their relationship to the work in its entirety . function includes the artist ’ s intended use ( s ) for the work and the actual use ( s ) of the work , which may change according to the context of audience , time , location , and culture . functions may be for utility , intercession , decoration , communication , and commemoration and may be spiritual , social , political , and/or personally expressive . materials ( or medium ) include raw ingredients ( such as pigment , wood , and limestone ) , compounds ( such as textile , ceramic , and ink ) , and components ( such as beads , paper , and performance ) used to create a work of art . specific materials have inherent properties ( e.g. , pliability , fragility , and permanence ) and tend to accrue cultural value ( e.g. , the value of gold or feathers due to relative rarity or exoticism ) . presentation is the display , enactment , and/or appearance of a work of art . response is the reaction of a person or population to the experience generated by a work of art . responses from an audience to a work of art may be physical , perceptual , spiritual , intellectual , and/or emotional . style is a combination of unique and defining features that can reflect the historical period , geographic location , cultural context , and individual hand of the artist . techniques include art-making processes , tools , and technologies that accommodate and/or overcome material properties . techniques range from simple to complex and easy to difficult , and may be practiced by one artist or may necessitate a group effort . a work of art is created by the artist ’ s deliberate manipulation of materials and techniques to produce purposeful form and content , which may be architecture , an object , an act , and/or an event . a work of art may be two- , three- , or four-dimensional ( time-based and performative ) . © 2013 the college board
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aesthetic refers to a type of human experience that combines perception , feeling , meaning making , and appreciation of qualities of produced and/or manipulated objects , acts , and events of daily life . aesthetic experience motivates behavior and creates categories through which our experiences of the world can be organized .
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what is overlapping in an artwork ?
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aesthetic refers to a type of human experience that combines perception , feeling , meaning making , and appreciation of qualities of produced and/or manipulated objects , acts , and events of daily life . aesthetic experience motivates behavior and creates categories through which our experiences of the world can be organized . artistic associations include self-defined groups , workshops , academies , and movements . artistic traditions are norms of artistic production and artistic products . artistic traditions are demonstrated through art-making processes ( utilization of materials and techniques , mode of display ) , through interactions between works of art and audience , and within form and/or content of a work of art . artistic changes are divergences from tradition in artistic choices demonstrated through art-making processes , through interactions between works of art and audience , and within form and/or content . tradition and change in form and content may be described in terms of style . audiences of a work of art are those who interact with the work as participants , facilitators , and/or observers . audience characteristics include gender , ethnicity , race , age , socioeconomic status , beliefs , and values . audience groups may be contemporaries , descendants , collectors , scholars , gallery/museum visitors , and other artists . content of a work of art consists of interacting , communicative elements of design , representation , and presentation within a work of art . content includes subject matter : visible imagery that may be formal depictions ( e.g. , minimalist or nonobjective works ) , representative depictions ( e.g. , portraiture and landscape ) , and/or symbolic depictions ( e.g. , emblems and logos ) . content may be narrative , symbolic , spiritual , historical , mythological , supernatural , and/or propagandistic ( e.g. , satirical and/or protest oriented ) . context includes original and subsequent historical and cultural milieu of a work of art . context includes information about the time , place , and culture in which a work of art was created , as well as information about when , where , and how subsequent audiences interacted with the work . the artist ’ s intended purpose for a work of art is contextual information , as is the chosen site for the work ( which may be public or private ) , as well as subsequent locations of the work . modes of display of a work of art can include associated paraphernalia ( e.g. , ceremonial objects and attire ) and multisensory stimuli ( e.g. , scent and sound ) . characteristics of the artist and audience—including aesthetic , intellectual , religious , political , social , and economic characteristics—are context . patronage , ownership of a work of art , and other power relationships are also aspects of context . contextual information includes audience response to a work of art . contextual information may be provided through records , reports , religious chronicles , personal reflections , manifestos , academic publications , mass media , sociological data , cultural studies , geographic data , artifacts , narrative and/or performance ( e.g. , oral , written , poetry , music , dance , dramatic productions ) , documentation , archaeology , and research . design elements are line , shape , color ( hue , value , saturation ) , texture , value ( shading ) , space , and form . design principles are balance/symmetry , rhythm/pattern , movement , harmony , contrast , emphasis , proportion/scale , and unity . form describes component materials and how they are employed to create physical and visual elements that coalesce into a work of art . form is investigated by applying design elements and principles to analyze the work ’ s fundamental visual components and their relationship to the work in its entirety . function includes the artist ’ s intended use ( s ) for the work and the actual use ( s ) of the work , which may change according to the context of audience , time , location , and culture . functions may be for utility , intercession , decoration , communication , and commemoration and may be spiritual , social , political , and/or personally expressive . materials ( or medium ) include raw ingredients ( such as pigment , wood , and limestone ) , compounds ( such as textile , ceramic , and ink ) , and components ( such as beads , paper , and performance ) used to create a work of art . specific materials have inherent properties ( e.g. , pliability , fragility , and permanence ) and tend to accrue cultural value ( e.g. , the value of gold or feathers due to relative rarity or exoticism ) . presentation is the display , enactment , and/or appearance of a work of art . response is the reaction of a person or population to the experience generated by a work of art . responses from an audience to a work of art may be physical , perceptual , spiritual , intellectual , and/or emotional . style is a combination of unique and defining features that can reflect the historical period , geographic location , cultural context , and individual hand of the artist . techniques include art-making processes , tools , and technologies that accommodate and/or overcome material properties . techniques range from simple to complex and easy to difficult , and may be practiced by one artist or may necessitate a group effort . a work of art is created by the artist ’ s deliberate manipulation of materials and techniques to produce purposeful form and content , which may be architecture , an object , an act , and/or an event . a work of art may be two- , three- , or four-dimensional ( time-based and performative ) . © 2013 the college board
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artistic traditions are norms of artistic production and artistic products . artistic traditions are demonstrated through art-making processes ( utilization of materials and techniques , mode of display ) , through interactions between works of art and audience , and within form and/or content of a work of art . artistic changes are divergences from tradition in artistic choices demonstrated through art-making processes , through interactions between works of art and audience , and within form and/or content .
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how doe shadowing effect art ?
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aesthetic refers to a type of human experience that combines perception , feeling , meaning making , and appreciation of qualities of produced and/or manipulated objects , acts , and events of daily life . aesthetic experience motivates behavior and creates categories through which our experiences of the world can be organized . artistic associations include self-defined groups , workshops , academies , and movements . artistic traditions are norms of artistic production and artistic products . artistic traditions are demonstrated through art-making processes ( utilization of materials and techniques , mode of display ) , through interactions between works of art and audience , and within form and/or content of a work of art . artistic changes are divergences from tradition in artistic choices demonstrated through art-making processes , through interactions between works of art and audience , and within form and/or content . tradition and change in form and content may be described in terms of style . audiences of a work of art are those who interact with the work as participants , facilitators , and/or observers . audience characteristics include gender , ethnicity , race , age , socioeconomic status , beliefs , and values . audience groups may be contemporaries , descendants , collectors , scholars , gallery/museum visitors , and other artists . content of a work of art consists of interacting , communicative elements of design , representation , and presentation within a work of art . content includes subject matter : visible imagery that may be formal depictions ( e.g. , minimalist or nonobjective works ) , representative depictions ( e.g. , portraiture and landscape ) , and/or symbolic depictions ( e.g. , emblems and logos ) . content may be narrative , symbolic , spiritual , historical , mythological , supernatural , and/or propagandistic ( e.g. , satirical and/or protest oriented ) . context includes original and subsequent historical and cultural milieu of a work of art . context includes information about the time , place , and culture in which a work of art was created , as well as information about when , where , and how subsequent audiences interacted with the work . the artist ’ s intended purpose for a work of art is contextual information , as is the chosen site for the work ( which may be public or private ) , as well as subsequent locations of the work . modes of display of a work of art can include associated paraphernalia ( e.g. , ceremonial objects and attire ) and multisensory stimuli ( e.g. , scent and sound ) . characteristics of the artist and audience—including aesthetic , intellectual , religious , political , social , and economic characteristics—are context . patronage , ownership of a work of art , and other power relationships are also aspects of context . contextual information includes audience response to a work of art . contextual information may be provided through records , reports , religious chronicles , personal reflections , manifestos , academic publications , mass media , sociological data , cultural studies , geographic data , artifacts , narrative and/or performance ( e.g. , oral , written , poetry , music , dance , dramatic productions ) , documentation , archaeology , and research . design elements are line , shape , color ( hue , value , saturation ) , texture , value ( shading ) , space , and form . design principles are balance/symmetry , rhythm/pattern , movement , harmony , contrast , emphasis , proportion/scale , and unity . form describes component materials and how they are employed to create physical and visual elements that coalesce into a work of art . form is investigated by applying design elements and principles to analyze the work ’ s fundamental visual components and their relationship to the work in its entirety . function includes the artist ’ s intended use ( s ) for the work and the actual use ( s ) of the work , which may change according to the context of audience , time , location , and culture . functions may be for utility , intercession , decoration , communication , and commemoration and may be spiritual , social , political , and/or personally expressive . materials ( or medium ) include raw ingredients ( such as pigment , wood , and limestone ) , compounds ( such as textile , ceramic , and ink ) , and components ( such as beads , paper , and performance ) used to create a work of art . specific materials have inherent properties ( e.g. , pliability , fragility , and permanence ) and tend to accrue cultural value ( e.g. , the value of gold or feathers due to relative rarity or exoticism ) . presentation is the display , enactment , and/or appearance of a work of art . response is the reaction of a person or population to the experience generated by a work of art . responses from an audience to a work of art may be physical , perceptual , spiritual , intellectual , and/or emotional . style is a combination of unique and defining features that can reflect the historical period , geographic location , cultural context , and individual hand of the artist . techniques include art-making processes , tools , and technologies that accommodate and/or overcome material properties . techniques range from simple to complex and easy to difficult , and may be practiced by one artist or may necessitate a group effort . a work of art is created by the artist ’ s deliberate manipulation of materials and techniques to produce purposeful form and content , which may be architecture , an object , an act , and/or an event . a work of art may be two- , three- , or four-dimensional ( time-based and performative ) . © 2013 the college board
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artistic traditions are norms of artistic production and artistic products . artistic traditions are demonstrated through art-making processes ( utilization of materials and techniques , mode of display ) , through interactions between works of art and audience , and within form and/or content of a work of art . artistic changes are divergences from tradition in artistic choices demonstrated through art-making processes , through interactions between works of art and audience , and within form and/or content .
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can art /art culture survive with out art associations ?
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aesthetic refers to a type of human experience that combines perception , feeling , meaning making , and appreciation of qualities of produced and/or manipulated objects , acts , and events of daily life . aesthetic experience motivates behavior and creates categories through which our experiences of the world can be organized . artistic associations include self-defined groups , workshops , academies , and movements . artistic traditions are norms of artistic production and artistic products . artistic traditions are demonstrated through art-making processes ( utilization of materials and techniques , mode of display ) , through interactions between works of art and audience , and within form and/or content of a work of art . artistic changes are divergences from tradition in artistic choices demonstrated through art-making processes , through interactions between works of art and audience , and within form and/or content . tradition and change in form and content may be described in terms of style . audiences of a work of art are those who interact with the work as participants , facilitators , and/or observers . audience characteristics include gender , ethnicity , race , age , socioeconomic status , beliefs , and values . audience groups may be contemporaries , descendants , collectors , scholars , gallery/museum visitors , and other artists . content of a work of art consists of interacting , communicative elements of design , representation , and presentation within a work of art . content includes subject matter : visible imagery that may be formal depictions ( e.g. , minimalist or nonobjective works ) , representative depictions ( e.g. , portraiture and landscape ) , and/or symbolic depictions ( e.g. , emblems and logos ) . content may be narrative , symbolic , spiritual , historical , mythological , supernatural , and/or propagandistic ( e.g. , satirical and/or protest oriented ) . context includes original and subsequent historical and cultural milieu of a work of art . context includes information about the time , place , and culture in which a work of art was created , as well as information about when , where , and how subsequent audiences interacted with the work . the artist ’ s intended purpose for a work of art is contextual information , as is the chosen site for the work ( which may be public or private ) , as well as subsequent locations of the work . modes of display of a work of art can include associated paraphernalia ( e.g. , ceremonial objects and attire ) and multisensory stimuli ( e.g. , scent and sound ) . characteristics of the artist and audience—including aesthetic , intellectual , religious , political , social , and economic characteristics—are context . patronage , ownership of a work of art , and other power relationships are also aspects of context . contextual information includes audience response to a work of art . contextual information may be provided through records , reports , religious chronicles , personal reflections , manifestos , academic publications , mass media , sociological data , cultural studies , geographic data , artifacts , narrative and/or performance ( e.g. , oral , written , poetry , music , dance , dramatic productions ) , documentation , archaeology , and research . design elements are line , shape , color ( hue , value , saturation ) , texture , value ( shading ) , space , and form . design principles are balance/symmetry , rhythm/pattern , movement , harmony , contrast , emphasis , proportion/scale , and unity . form describes component materials and how they are employed to create physical and visual elements that coalesce into a work of art . form is investigated by applying design elements and principles to analyze the work ’ s fundamental visual components and their relationship to the work in its entirety . function includes the artist ’ s intended use ( s ) for the work and the actual use ( s ) of the work , which may change according to the context of audience , time , location , and culture . functions may be for utility , intercession , decoration , communication , and commemoration and may be spiritual , social , political , and/or personally expressive . materials ( or medium ) include raw ingredients ( such as pigment , wood , and limestone ) , compounds ( such as textile , ceramic , and ink ) , and components ( such as beads , paper , and performance ) used to create a work of art . specific materials have inherent properties ( e.g. , pliability , fragility , and permanence ) and tend to accrue cultural value ( e.g. , the value of gold or feathers due to relative rarity or exoticism ) . presentation is the display , enactment , and/or appearance of a work of art . response is the reaction of a person or population to the experience generated by a work of art . responses from an audience to a work of art may be physical , perceptual , spiritual , intellectual , and/or emotional . style is a combination of unique and defining features that can reflect the historical period , geographic location , cultural context , and individual hand of the artist . techniques include art-making processes , tools , and technologies that accommodate and/or overcome material properties . techniques range from simple to complex and easy to difficult , and may be practiced by one artist or may necessitate a group effort . a work of art is created by the artist ’ s deliberate manipulation of materials and techniques to produce purposeful form and content , which may be architecture , an object , an act , and/or an event . a work of art may be two- , three- , or four-dimensional ( time-based and performative ) . © 2013 the college board
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style is a combination of unique and defining features that can reflect the historical period , geographic location , cultural context , and individual hand of the artist . techniques include art-making processes , tools , and technologies that accommodate and/or overcome material properties . techniques range from simple to complex and easy to difficult , and may be practiced by one artist or may necessitate a group effort .
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what are art making process tools ?
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aesthetic refers to a type of human experience that combines perception , feeling , meaning making , and appreciation of qualities of produced and/or manipulated objects , acts , and events of daily life . aesthetic experience motivates behavior and creates categories through which our experiences of the world can be organized . artistic associations include self-defined groups , workshops , academies , and movements . artistic traditions are norms of artistic production and artistic products . artistic traditions are demonstrated through art-making processes ( utilization of materials and techniques , mode of display ) , through interactions between works of art and audience , and within form and/or content of a work of art . artistic changes are divergences from tradition in artistic choices demonstrated through art-making processes , through interactions between works of art and audience , and within form and/or content . tradition and change in form and content may be described in terms of style . audiences of a work of art are those who interact with the work as participants , facilitators , and/or observers . audience characteristics include gender , ethnicity , race , age , socioeconomic status , beliefs , and values . audience groups may be contemporaries , descendants , collectors , scholars , gallery/museum visitors , and other artists . content of a work of art consists of interacting , communicative elements of design , representation , and presentation within a work of art . content includes subject matter : visible imagery that may be formal depictions ( e.g. , minimalist or nonobjective works ) , representative depictions ( e.g. , portraiture and landscape ) , and/or symbolic depictions ( e.g. , emblems and logos ) . content may be narrative , symbolic , spiritual , historical , mythological , supernatural , and/or propagandistic ( e.g. , satirical and/or protest oriented ) . context includes original and subsequent historical and cultural milieu of a work of art . context includes information about the time , place , and culture in which a work of art was created , as well as information about when , where , and how subsequent audiences interacted with the work . the artist ’ s intended purpose for a work of art is contextual information , as is the chosen site for the work ( which may be public or private ) , as well as subsequent locations of the work . modes of display of a work of art can include associated paraphernalia ( e.g. , ceremonial objects and attire ) and multisensory stimuli ( e.g. , scent and sound ) . characteristics of the artist and audience—including aesthetic , intellectual , religious , political , social , and economic characteristics—are context . patronage , ownership of a work of art , and other power relationships are also aspects of context . contextual information includes audience response to a work of art . contextual information may be provided through records , reports , religious chronicles , personal reflections , manifestos , academic publications , mass media , sociological data , cultural studies , geographic data , artifacts , narrative and/or performance ( e.g. , oral , written , poetry , music , dance , dramatic productions ) , documentation , archaeology , and research . design elements are line , shape , color ( hue , value , saturation ) , texture , value ( shading ) , space , and form . design principles are balance/symmetry , rhythm/pattern , movement , harmony , contrast , emphasis , proportion/scale , and unity . form describes component materials and how they are employed to create physical and visual elements that coalesce into a work of art . form is investigated by applying design elements and principles to analyze the work ’ s fundamental visual components and their relationship to the work in its entirety . function includes the artist ’ s intended use ( s ) for the work and the actual use ( s ) of the work , which may change according to the context of audience , time , location , and culture . functions may be for utility , intercession , decoration , communication , and commemoration and may be spiritual , social , political , and/or personally expressive . materials ( or medium ) include raw ingredients ( such as pigment , wood , and limestone ) , compounds ( such as textile , ceramic , and ink ) , and components ( such as beads , paper , and performance ) used to create a work of art . specific materials have inherent properties ( e.g. , pliability , fragility , and permanence ) and tend to accrue cultural value ( e.g. , the value of gold or feathers due to relative rarity or exoticism ) . presentation is the display , enactment , and/or appearance of a work of art . response is the reaction of a person or population to the experience generated by a work of art . responses from an audience to a work of art may be physical , perceptual , spiritual , intellectual , and/or emotional . style is a combination of unique and defining features that can reflect the historical period , geographic location , cultural context , and individual hand of the artist . techniques include art-making processes , tools , and technologies that accommodate and/or overcome material properties . techniques range from simple to complex and easy to difficult , and may be practiced by one artist or may necessitate a group effort . a work of art is created by the artist ’ s deliberate manipulation of materials and techniques to produce purposeful form and content , which may be architecture , an object , an act , and/or an event . a work of art may be two- , three- , or four-dimensional ( time-based and performative ) . © 2013 the college board
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aesthetic refers to a type of human experience that combines perception , feeling , meaning making , and appreciation of qualities of produced and/or manipulated objects , acts , and events of daily life . aesthetic experience motivates behavior and creates categories through which our experiences of the world can be organized .
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what are the key factors to remember ?
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for most of us , architecture is easy to take for granted . its everywhere in our daily lives—sometimes elegant , other times shabby , but generally ubiquitous . how often do we stop to examine and contemplate its form and style ? stopping for that contemplation offers not only the opportunity to understand one ’ s daily surroundings , but also to appreciate the connection that exists between architectural forms in our own time and those from the past . architectural tradition and design has the ability to link disparate cultures together over time and space—and this is certainly true of the legacy of architectural forms created by the ancient greeks . greek architecture refers to the architecture of the greek-speaking peoples who inhabited the greek mainland and the peloponnese , the islands of the aegean sea , the greek colonies in ionia ( coastal asia minor ) , and magna graecia ( greek colonies in italy and sicily ) . greek architecture stretches from c. 900 b.c.e . to the first century c.e . ( with the earliest extant stone architecture dating to the seventh century b.c.e . ) . greek architecture influenced roman architecture and architects in profound ways , such that roman imperial architecture adopts and incorporates many greek elements into its own practice . an overview of basic building typologies demonstrates the range and diversity of greek architecture . temple the most recognizably “ greek ” structure is the temple ( even though the architecture of greek temples is actually quite diverse ) . the greeks referred to temples with the term ὁ ναός ( ho naós ) meaning `` dwelling ; '' temple derives from the latin term , templum . the earliest shrines were built to honor divinities and were made from materials such as a wood and mud brick—materials that typically do n't survive very long . the basic form of the naos emerges as early as the tenth century b.c.e . as a simple , rectangular room with projecting walls ( antae ) that created a shallow porch . this basic form remained unchanged in its concept for centuries . in the eighth century b.c.e . greek architecture begins to make the move from ephemeral materials ( wood , mud brick , thatch ) to permanent materials ( namely , stone ) . during the archaic period the tenets of the doric order of architecture in the greek mainland became firmly established , leading to a wave of monumental temple building during the sixth and fifth centuries b.c.e . greek city-states invested substantial resources in temple building—as they competed with each other not just in strategic and economic terms , but also in their architecture . for example , athens devoted enormous resources to the construction of the acropolis in the 5th century b.c.e.—in part so that athenians could be confident that the temples built to honor their gods surpassed anything that their rival states could offer . the multi-phase architectural development of sanctuaries such as that of hera on the island of samos demonstrate not only the change that occurred in construction techniques over time but also how the greeks re-used sacred spaces—with the later phases built directly atop the preceding ones . perhaps the fullest , and most famous , expression of classical greek temple architecture is the periclean parthenon of athens—a doric order structure , the parthenon represents the maturity of the greek classical form . greek temples are often categorized in terms of their ground plan and the way in which the columns are arranged . a prostyle temple is a temple that has columns only at the front , while an amphiprostyle temple has columns at the front and the rear . temples with a peripteral arrangement ( from the greek πτερον ( pteron ) meaning `` wing ) have a single line of columns arranged all around the exterior of the temple building . dipteral temples simply have a double row of columns surrounding the building . one of the more unusual plans is the tholos , a temple with a circular ground plan ; famous examples are attested at the sanctuary of apollo in delphi and the sanctuary of asclepius at epidauros . stoa stoa ( στοά ) is a greek architectural term that describes a covered walkway or colonnade that was usually designed for public use . early examples , often employing the doric order , were usually composed of a single level , although later examples ( hellenistic and roman ) came to be two-story freestanding structures . these later examples allowed interior space for shops or other rooms and often incorporated the ionic order for interior colonnades . greek city planners came to prefer the stoa as a device for framing the agora ( public market place ) of a city or town . the south stoa constructed as part of the sanctuary of hera on the island of samos ( c. 700-550 b.c.e . ) numbers among the earliest examples of the stoa in greek architecture . many cities , particularly athens and corinth , came to have elaborate and famous stoas . in athens the famous stoa poikile ( “ painted stoa ” ) , c. fifth century b.c.e. , housed paintings of famous greek military exploits including the battle of marathon , while the stoa basileios ( “ royal stoa ” ) , c. fifth century b.c.e. , was the seat of a chief civic official ( archon basileios ) . later , through the patronage of the kings of pergamon , the athenian agora was augmented by the famed stoa of attalos ( c. 159-138 b.c.e . ) which was recently rebuilt according to the ancient specifications and now houses the archaeological museum for the athenian agora itself ( see image above ) . at corinth the stoa persisted as an architectural type well into the roman period ; the south stoa there ( above ) , c. 150 c.e. , shows the continued utility of this building design for framing civic space . from the hellenistic period onwards the stoa also lent its name to a philosophical school , as zeno of citium ( c. 334-262 b.c.e . ) originally taught his stoic philosophy in the stoa poikile of athens . theater the greek theater was a large , open-air structure used for dramatic performance . theaters often took advantage of hillsides and naturally sloping terrain and , in general , utilized the panoramic landscape as the backdrop to the stage itself . the greek theater is composed of the seating area ( theatron ) , a circular space for the chorus to perform ( orchestra ) , and the stage ( skene ) . tiered seats in the theatron provided space for spectators . two side aisles ( parados , pl . paradoi ) provided access to the orchestra . the greek theater inspired the roman version of the theater directly , although the romans introduced some modifications to the concept of theater architecture . in many cases the romans converted pre-existing greek theaters to conform to their own architectural ideals , as is evident in the theater of dionysos on the slopes of the athenian acropolis . since theatrical performances were often linked to sacred festivals , it is not uncommon to find theaters associated directly with sanctuaries . bouleuterion the bouleuterion ( βουλευτήριον ) was an important civic building in a greek city , as it was the meeting place of the boule ( citizen council ) of the city . these select representatives assembled to handle public affairs and represent the citizenry of the polis ( in ancient athens the boule was comprised of 500 members ) . the bouleuterion generally was a covered , rectilinear building with stepped seating surrounding a central speaker ’ s well in which an altar was placed . the city of priène has a particularly well-preserved example of this civic structure as does the city of miletus . house greek houses of the archaic and classical periods were relatively simple in design . houses usually were centered on a courtyard that would have been the scene for various ritual activities ; the courtyard also provided natural light for the often small houses . the ground floor rooms would have included kitchen and storage rooms , perhaps an animal pen and a latrine ; the chief room was the andron— site of the male-dominated drinking party ( symposion ) . the quarters for women and children ( gynaikeion ) could be located on the second level ( if present ) and were , in any case , segregated from the mens ’ area . it was not uncommon for houses to be attached to workshops or shops . the houses excavated in the southwest part of the athenian agora had walls of mud brick that rested on stone socles and tiled roofs , with floors of beaten clay . the city of olynthus in chalcidice , greece , destroyed by military action in 348 b.c.e. , preserves many well-appointed courtyard houses arranged within the hippodamian grid-plan of the city . house a vii 4 had a large cobbled courtyard that was used for domestic industry . while some rooms were fairly plain , with earthen floors , the andron was the most well-appointed room of the house . fortifications the mycenaean fortifications of bronze age greece ( c. 1300 b.c.e . ) are particularly well known—the megalithic architecture ( also referred to as cyclopean because of the use of enormous stones ) represents a trend in bronze age architecture . while these massive bronze age walls are difficult to best , first millennium b.c.e . greece also shows evidence for stone built fortification walls . in attika ( the territory of athens ) , a series of classical and hellenistic walls built in ashlar masonry ( squared masonry blocks ) have been studied as a potential system of border defenses . at palairos in epirus ( greece ) the massive fortifications enclose a high citadel that occupies imposing terrain . stadium , gymnasium , and palaestra the greek stadium ( derived from stadion , a greek measurement equivalent to c. 578 feet or 176 meters ) was the location of foot races held as part of sacred games ; these structures are often found in the context of sanctuaries , as in the case of the panhellenic sanctuaries at olympia and epidauros . long and narrow , with a horseshoe shape , the stadium occupied reasonably flat terrain . the gymnasium ( from the greek term gymnós meaning `` naked '' ) was a training center for athletes who participated in public games . this facility tended to include areas for both training and storage . the palaestra ( παλαίστρα ) was an exercise facility originally connected with the training of wrestlers . these complexes were generally rectilinear in plan , with a colonnade framing a central , open space . altar since blood sacrifice was a key component of greek ritual practice , an altar was essential for these purposes . while altars did not necessarily need to be architecturalized , they could be and , in some cases , they assumed a monumental scale . the third century b.c.e . altar of hieron ii at syracuse , sicily , provides one such example . at c. 196 meters in length and c. 11 m in height the massive altar was reported to be capable of hosting the simultaneous sacrifice of 450 bulls ( diodorus siculus history 11.72.2 ) . another spectacular altar is the altar of zeus from pergamon , built during the first half of the second century b.c.e . the altar itself is screened by a monumental enclosure decorated with sculpture ; the monument measures c. 35.64 by 33.4 meters . the altar is best known for its program of relief sculpture that depict a gigantomachy ( battle between the olympian gods and the giants ) that is presented as an allegory for the military conquests of the kings of pergamon . despite its monumental scale and lavish decoration , the pergamon altar preserves the basic and necessary features of the greek altar : it is frontal and approached by stairs and is open to the air—to allow not only for the blood sacrifice itself but also for the burning of the thigh bones and fat as an offering to the gods . fountain house the fountain house is a public building that provides access to clean drinking water and at which water jars and containers could be filled . the southeast fountain house in the athenian agora ( c. 530 b.c.e . ) provides an example of this tendency to position fountain houses and their dependable supply of clean drinking water close to civic spaces like the agora . gathering water was seen as a woman ’ s task and , as such , it offered the often isolated women a chance to socialize with others while collecting water . fountain house scenes are common on ceramic water jars ( hydriai ) , as is the case for a black-figured hydria ( c. 525-500 b.c.e . ) found in an etruscan tomb in vulci that is now in the british museum legacy the architecture of ancient greece influenced ancient roman architecture , and became the architectural vernacular employed in the expansive hellenistic world created in the wake of the conquests of alexander the great . greek architectural forms became implanted so deeply in the roman architectural mindset that they endured throughout antiquity , only to then be re-discovered in the renaissance and especially from the mid-eighteenth century onwards as a feature of the neo-classical movement . this durable legacy helps to explain why the ancient greek architectural orders and the tenets of greek design are still so prevalent—and visible—in our post-modern world . essay by dr. jeffrey a. becker additional resources : athenian agora excavations j. m. camp , the athenian agora : a short guide to the excavations ( american school of classical studies at athens ) architecture in ancient greece on the metropolitan museum of art 's heilbrunn timeline of art history b . a. ault and l. nevett , ancient greek houses and households : chronological , regional , and social diversity ( philadelphia : university of pennsylvania press , 2005 ) . n. cahill , household and city organization at olynthus ( new haven : yale university press , 2001 ) . j. j. coulton , the architectural development of the greek stoa ( oxford : clarendon press , 1976 ) . j. j. coulton , ancient greek architects at work : problems of structure and design ( ithaca ny : cornell university press , 1982 ) . w. b. dinsmoor , the architecture of greece : an account of its historic development 3rd ed . ( london : batsford , 1950 ) . marie-christine hellmann , l ’ architecture grecque 3 vol . ( paris : picard , 2002-2010 ) . m. korres , stones of the parthenon ( los angeles : j. paul getty museum , 2000 ) . a. w. lawrence , greek architecture 5th ed . ( new haven : yale university press , 1996 ) . c. g. malacrino , constructing the ancient world : architectural techniques of the greeks and romans ( los angeles : j. paul getty museum , 2010 ) . a. mazarakis ainian , from rulers ' dwellings to temples : architecture , religion and society in early iron age greece ( 1100-700 b.c . ) ( jonsered : p. åströms förlag , 1997 ) . l. nevett , house and society in the ancient greek world ( cambridge : cambridge university press , 1999 ) . j. ober , fortress attica : defense of the athenian land frontier , 404-322 b.c . ( leiden : e. j. brill , 1985 ) . d. s. robertson , greek and roman architecture 2nd ed . ( cambridge : cambridge university press , 1969 ) . j. n. travlos , pictorial dictionary of ancient athens ( new york : praeger , 1971 ) . f. e. winter , greek fortifications ( toronto : university of toronto press , 1971 ) . f. e. winter , studies in hellenistic architecture ( toronto : university of toronto press , 2006 ) . w. wrede , attische mauern ( athens : deutsches archäologisches institut , 1933 ) . r. e. wycherley , the stones of athens ( princeton : princeton university press , 1978 ) .
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tiered seats in the theatron provided space for spectators . two side aisles ( parados , pl . paradoi ) provided access to the orchestra .
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is there any logic behind the fact that these two rather different structures share the same name ?
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for most of us , architecture is easy to take for granted . its everywhere in our daily lives—sometimes elegant , other times shabby , but generally ubiquitous . how often do we stop to examine and contemplate its form and style ? stopping for that contemplation offers not only the opportunity to understand one ’ s daily surroundings , but also to appreciate the connection that exists between architectural forms in our own time and those from the past . architectural tradition and design has the ability to link disparate cultures together over time and space—and this is certainly true of the legacy of architectural forms created by the ancient greeks . greek architecture refers to the architecture of the greek-speaking peoples who inhabited the greek mainland and the peloponnese , the islands of the aegean sea , the greek colonies in ionia ( coastal asia minor ) , and magna graecia ( greek colonies in italy and sicily ) . greek architecture stretches from c. 900 b.c.e . to the first century c.e . ( with the earliest extant stone architecture dating to the seventh century b.c.e . ) . greek architecture influenced roman architecture and architects in profound ways , such that roman imperial architecture adopts and incorporates many greek elements into its own practice . an overview of basic building typologies demonstrates the range and diversity of greek architecture . temple the most recognizably “ greek ” structure is the temple ( even though the architecture of greek temples is actually quite diverse ) . the greeks referred to temples with the term ὁ ναός ( ho naós ) meaning `` dwelling ; '' temple derives from the latin term , templum . the earliest shrines were built to honor divinities and were made from materials such as a wood and mud brick—materials that typically do n't survive very long . the basic form of the naos emerges as early as the tenth century b.c.e . as a simple , rectangular room with projecting walls ( antae ) that created a shallow porch . this basic form remained unchanged in its concept for centuries . in the eighth century b.c.e . greek architecture begins to make the move from ephemeral materials ( wood , mud brick , thatch ) to permanent materials ( namely , stone ) . during the archaic period the tenets of the doric order of architecture in the greek mainland became firmly established , leading to a wave of monumental temple building during the sixth and fifth centuries b.c.e . greek city-states invested substantial resources in temple building—as they competed with each other not just in strategic and economic terms , but also in their architecture . for example , athens devoted enormous resources to the construction of the acropolis in the 5th century b.c.e.—in part so that athenians could be confident that the temples built to honor their gods surpassed anything that their rival states could offer . the multi-phase architectural development of sanctuaries such as that of hera on the island of samos demonstrate not only the change that occurred in construction techniques over time but also how the greeks re-used sacred spaces—with the later phases built directly atop the preceding ones . perhaps the fullest , and most famous , expression of classical greek temple architecture is the periclean parthenon of athens—a doric order structure , the parthenon represents the maturity of the greek classical form . greek temples are often categorized in terms of their ground plan and the way in which the columns are arranged . a prostyle temple is a temple that has columns only at the front , while an amphiprostyle temple has columns at the front and the rear . temples with a peripteral arrangement ( from the greek πτερον ( pteron ) meaning `` wing ) have a single line of columns arranged all around the exterior of the temple building . dipteral temples simply have a double row of columns surrounding the building . one of the more unusual plans is the tholos , a temple with a circular ground plan ; famous examples are attested at the sanctuary of apollo in delphi and the sanctuary of asclepius at epidauros . stoa stoa ( στοά ) is a greek architectural term that describes a covered walkway or colonnade that was usually designed for public use . early examples , often employing the doric order , were usually composed of a single level , although later examples ( hellenistic and roman ) came to be two-story freestanding structures . these later examples allowed interior space for shops or other rooms and often incorporated the ionic order for interior colonnades . greek city planners came to prefer the stoa as a device for framing the agora ( public market place ) of a city or town . the south stoa constructed as part of the sanctuary of hera on the island of samos ( c. 700-550 b.c.e . ) numbers among the earliest examples of the stoa in greek architecture . many cities , particularly athens and corinth , came to have elaborate and famous stoas . in athens the famous stoa poikile ( “ painted stoa ” ) , c. fifth century b.c.e. , housed paintings of famous greek military exploits including the battle of marathon , while the stoa basileios ( “ royal stoa ” ) , c. fifth century b.c.e. , was the seat of a chief civic official ( archon basileios ) . later , through the patronage of the kings of pergamon , the athenian agora was augmented by the famed stoa of attalos ( c. 159-138 b.c.e . ) which was recently rebuilt according to the ancient specifications and now houses the archaeological museum for the athenian agora itself ( see image above ) . at corinth the stoa persisted as an architectural type well into the roman period ; the south stoa there ( above ) , c. 150 c.e. , shows the continued utility of this building design for framing civic space . from the hellenistic period onwards the stoa also lent its name to a philosophical school , as zeno of citium ( c. 334-262 b.c.e . ) originally taught his stoic philosophy in the stoa poikile of athens . theater the greek theater was a large , open-air structure used for dramatic performance . theaters often took advantage of hillsides and naturally sloping terrain and , in general , utilized the panoramic landscape as the backdrop to the stage itself . the greek theater is composed of the seating area ( theatron ) , a circular space for the chorus to perform ( orchestra ) , and the stage ( skene ) . tiered seats in the theatron provided space for spectators . two side aisles ( parados , pl . paradoi ) provided access to the orchestra . the greek theater inspired the roman version of the theater directly , although the romans introduced some modifications to the concept of theater architecture . in many cases the romans converted pre-existing greek theaters to conform to their own architectural ideals , as is evident in the theater of dionysos on the slopes of the athenian acropolis . since theatrical performances were often linked to sacred festivals , it is not uncommon to find theaters associated directly with sanctuaries . bouleuterion the bouleuterion ( βουλευτήριον ) was an important civic building in a greek city , as it was the meeting place of the boule ( citizen council ) of the city . these select representatives assembled to handle public affairs and represent the citizenry of the polis ( in ancient athens the boule was comprised of 500 members ) . the bouleuterion generally was a covered , rectilinear building with stepped seating surrounding a central speaker ’ s well in which an altar was placed . the city of priène has a particularly well-preserved example of this civic structure as does the city of miletus . house greek houses of the archaic and classical periods were relatively simple in design . houses usually were centered on a courtyard that would have been the scene for various ritual activities ; the courtyard also provided natural light for the often small houses . the ground floor rooms would have included kitchen and storage rooms , perhaps an animal pen and a latrine ; the chief room was the andron— site of the male-dominated drinking party ( symposion ) . the quarters for women and children ( gynaikeion ) could be located on the second level ( if present ) and were , in any case , segregated from the mens ’ area . it was not uncommon for houses to be attached to workshops or shops . the houses excavated in the southwest part of the athenian agora had walls of mud brick that rested on stone socles and tiled roofs , with floors of beaten clay . the city of olynthus in chalcidice , greece , destroyed by military action in 348 b.c.e. , preserves many well-appointed courtyard houses arranged within the hippodamian grid-plan of the city . house a vii 4 had a large cobbled courtyard that was used for domestic industry . while some rooms were fairly plain , with earthen floors , the andron was the most well-appointed room of the house . fortifications the mycenaean fortifications of bronze age greece ( c. 1300 b.c.e . ) are particularly well known—the megalithic architecture ( also referred to as cyclopean because of the use of enormous stones ) represents a trend in bronze age architecture . while these massive bronze age walls are difficult to best , first millennium b.c.e . greece also shows evidence for stone built fortification walls . in attika ( the territory of athens ) , a series of classical and hellenistic walls built in ashlar masonry ( squared masonry blocks ) have been studied as a potential system of border defenses . at palairos in epirus ( greece ) the massive fortifications enclose a high citadel that occupies imposing terrain . stadium , gymnasium , and palaestra the greek stadium ( derived from stadion , a greek measurement equivalent to c. 578 feet or 176 meters ) was the location of foot races held as part of sacred games ; these structures are often found in the context of sanctuaries , as in the case of the panhellenic sanctuaries at olympia and epidauros . long and narrow , with a horseshoe shape , the stadium occupied reasonably flat terrain . the gymnasium ( from the greek term gymnós meaning `` naked '' ) was a training center for athletes who participated in public games . this facility tended to include areas for both training and storage . the palaestra ( παλαίστρα ) was an exercise facility originally connected with the training of wrestlers . these complexes were generally rectilinear in plan , with a colonnade framing a central , open space . altar since blood sacrifice was a key component of greek ritual practice , an altar was essential for these purposes . while altars did not necessarily need to be architecturalized , they could be and , in some cases , they assumed a monumental scale . the third century b.c.e . altar of hieron ii at syracuse , sicily , provides one such example . at c. 196 meters in length and c. 11 m in height the massive altar was reported to be capable of hosting the simultaneous sacrifice of 450 bulls ( diodorus siculus history 11.72.2 ) . another spectacular altar is the altar of zeus from pergamon , built during the first half of the second century b.c.e . the altar itself is screened by a monumental enclosure decorated with sculpture ; the monument measures c. 35.64 by 33.4 meters . the altar is best known for its program of relief sculpture that depict a gigantomachy ( battle between the olympian gods and the giants ) that is presented as an allegory for the military conquests of the kings of pergamon . despite its monumental scale and lavish decoration , the pergamon altar preserves the basic and necessary features of the greek altar : it is frontal and approached by stairs and is open to the air—to allow not only for the blood sacrifice itself but also for the burning of the thigh bones and fat as an offering to the gods . fountain house the fountain house is a public building that provides access to clean drinking water and at which water jars and containers could be filled . the southeast fountain house in the athenian agora ( c. 530 b.c.e . ) provides an example of this tendency to position fountain houses and their dependable supply of clean drinking water close to civic spaces like the agora . gathering water was seen as a woman ’ s task and , as such , it offered the often isolated women a chance to socialize with others while collecting water . fountain house scenes are common on ceramic water jars ( hydriai ) , as is the case for a black-figured hydria ( c. 525-500 b.c.e . ) found in an etruscan tomb in vulci that is now in the british museum legacy the architecture of ancient greece influenced ancient roman architecture , and became the architectural vernacular employed in the expansive hellenistic world created in the wake of the conquests of alexander the great . greek architectural forms became implanted so deeply in the roman architectural mindset that they endured throughout antiquity , only to then be re-discovered in the renaissance and especially from the mid-eighteenth century onwards as a feature of the neo-classical movement . this durable legacy helps to explain why the ancient greek architectural orders and the tenets of greek design are still so prevalent—and visible—in our post-modern world . essay by dr. jeffrey a. becker additional resources : athenian agora excavations j. m. camp , the athenian agora : a short guide to the excavations ( american school of classical studies at athens ) architecture in ancient greece on the metropolitan museum of art 's heilbrunn timeline of art history b . a. ault and l. nevett , ancient greek houses and households : chronological , regional , and social diversity ( philadelphia : university of pennsylvania press , 2005 ) . n. cahill , household and city organization at olynthus ( new haven : yale university press , 2001 ) . j. j. coulton , the architectural development of the greek stoa ( oxford : clarendon press , 1976 ) . j. j. coulton , ancient greek architects at work : problems of structure and design ( ithaca ny : cornell university press , 1982 ) . w. b. dinsmoor , the architecture of greece : an account of its historic development 3rd ed . ( london : batsford , 1950 ) . marie-christine hellmann , l ’ architecture grecque 3 vol . ( paris : picard , 2002-2010 ) . m. korres , stones of the parthenon ( los angeles : j. paul getty museum , 2000 ) . a. w. lawrence , greek architecture 5th ed . ( new haven : yale university press , 1996 ) . c. g. malacrino , constructing the ancient world : architectural techniques of the greeks and romans ( los angeles : j. paul getty museum , 2010 ) . a. mazarakis ainian , from rulers ' dwellings to temples : architecture , religion and society in early iron age greece ( 1100-700 b.c . ) ( jonsered : p. åströms förlag , 1997 ) . l. nevett , house and society in the ancient greek world ( cambridge : cambridge university press , 1999 ) . j. ober , fortress attica : defense of the athenian land frontier , 404-322 b.c . ( leiden : e. j. brill , 1985 ) . d. s. robertson , greek and roman architecture 2nd ed . ( cambridge : cambridge university press , 1969 ) . j. n. travlos , pictorial dictionary of ancient athens ( new york : praeger , 1971 ) . f. e. winter , greek fortifications ( toronto : university of toronto press , 1971 ) . f. e. winter , studies in hellenistic architecture ( toronto : university of toronto press , 2006 ) . w. wrede , attische mauern ( athens : deutsches archäologisches institut , 1933 ) . r. e. wycherley , the stones of athens ( princeton : princeton university press , 1978 ) .
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for most of us , architecture is easy to take for granted . its everywhere in our daily lives—sometimes elegant , other times shabby , but generally ubiquitous .
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`` ... i hope it was only animals being sacrificed ... no people right ?
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for most of us , architecture is easy to take for granted . its everywhere in our daily lives—sometimes elegant , other times shabby , but generally ubiquitous . how often do we stop to examine and contemplate its form and style ? stopping for that contemplation offers not only the opportunity to understand one ’ s daily surroundings , but also to appreciate the connection that exists between architectural forms in our own time and those from the past . architectural tradition and design has the ability to link disparate cultures together over time and space—and this is certainly true of the legacy of architectural forms created by the ancient greeks . greek architecture refers to the architecture of the greek-speaking peoples who inhabited the greek mainland and the peloponnese , the islands of the aegean sea , the greek colonies in ionia ( coastal asia minor ) , and magna graecia ( greek colonies in italy and sicily ) . greek architecture stretches from c. 900 b.c.e . to the first century c.e . ( with the earliest extant stone architecture dating to the seventh century b.c.e . ) . greek architecture influenced roman architecture and architects in profound ways , such that roman imperial architecture adopts and incorporates many greek elements into its own practice . an overview of basic building typologies demonstrates the range and diversity of greek architecture . temple the most recognizably “ greek ” structure is the temple ( even though the architecture of greek temples is actually quite diverse ) . the greeks referred to temples with the term ὁ ναός ( ho naós ) meaning `` dwelling ; '' temple derives from the latin term , templum . the earliest shrines were built to honor divinities and were made from materials such as a wood and mud brick—materials that typically do n't survive very long . the basic form of the naos emerges as early as the tenth century b.c.e . as a simple , rectangular room with projecting walls ( antae ) that created a shallow porch . this basic form remained unchanged in its concept for centuries . in the eighth century b.c.e . greek architecture begins to make the move from ephemeral materials ( wood , mud brick , thatch ) to permanent materials ( namely , stone ) . during the archaic period the tenets of the doric order of architecture in the greek mainland became firmly established , leading to a wave of monumental temple building during the sixth and fifth centuries b.c.e . greek city-states invested substantial resources in temple building—as they competed with each other not just in strategic and economic terms , but also in their architecture . for example , athens devoted enormous resources to the construction of the acropolis in the 5th century b.c.e.—in part so that athenians could be confident that the temples built to honor their gods surpassed anything that their rival states could offer . the multi-phase architectural development of sanctuaries such as that of hera on the island of samos demonstrate not only the change that occurred in construction techniques over time but also how the greeks re-used sacred spaces—with the later phases built directly atop the preceding ones . perhaps the fullest , and most famous , expression of classical greek temple architecture is the periclean parthenon of athens—a doric order structure , the parthenon represents the maturity of the greek classical form . greek temples are often categorized in terms of their ground plan and the way in which the columns are arranged . a prostyle temple is a temple that has columns only at the front , while an amphiprostyle temple has columns at the front and the rear . temples with a peripteral arrangement ( from the greek πτερον ( pteron ) meaning `` wing ) have a single line of columns arranged all around the exterior of the temple building . dipteral temples simply have a double row of columns surrounding the building . one of the more unusual plans is the tholos , a temple with a circular ground plan ; famous examples are attested at the sanctuary of apollo in delphi and the sanctuary of asclepius at epidauros . stoa stoa ( στοά ) is a greek architectural term that describes a covered walkway or colonnade that was usually designed for public use . early examples , often employing the doric order , were usually composed of a single level , although later examples ( hellenistic and roman ) came to be two-story freestanding structures . these later examples allowed interior space for shops or other rooms and often incorporated the ionic order for interior colonnades . greek city planners came to prefer the stoa as a device for framing the agora ( public market place ) of a city or town . the south stoa constructed as part of the sanctuary of hera on the island of samos ( c. 700-550 b.c.e . ) numbers among the earliest examples of the stoa in greek architecture . many cities , particularly athens and corinth , came to have elaborate and famous stoas . in athens the famous stoa poikile ( “ painted stoa ” ) , c. fifth century b.c.e. , housed paintings of famous greek military exploits including the battle of marathon , while the stoa basileios ( “ royal stoa ” ) , c. fifth century b.c.e. , was the seat of a chief civic official ( archon basileios ) . later , through the patronage of the kings of pergamon , the athenian agora was augmented by the famed stoa of attalos ( c. 159-138 b.c.e . ) which was recently rebuilt according to the ancient specifications and now houses the archaeological museum for the athenian agora itself ( see image above ) . at corinth the stoa persisted as an architectural type well into the roman period ; the south stoa there ( above ) , c. 150 c.e. , shows the continued utility of this building design for framing civic space . from the hellenistic period onwards the stoa also lent its name to a philosophical school , as zeno of citium ( c. 334-262 b.c.e . ) originally taught his stoic philosophy in the stoa poikile of athens . theater the greek theater was a large , open-air structure used for dramatic performance . theaters often took advantage of hillsides and naturally sloping terrain and , in general , utilized the panoramic landscape as the backdrop to the stage itself . the greek theater is composed of the seating area ( theatron ) , a circular space for the chorus to perform ( orchestra ) , and the stage ( skene ) . tiered seats in the theatron provided space for spectators . two side aisles ( parados , pl . paradoi ) provided access to the orchestra . the greek theater inspired the roman version of the theater directly , although the romans introduced some modifications to the concept of theater architecture . in many cases the romans converted pre-existing greek theaters to conform to their own architectural ideals , as is evident in the theater of dionysos on the slopes of the athenian acropolis . since theatrical performances were often linked to sacred festivals , it is not uncommon to find theaters associated directly with sanctuaries . bouleuterion the bouleuterion ( βουλευτήριον ) was an important civic building in a greek city , as it was the meeting place of the boule ( citizen council ) of the city . these select representatives assembled to handle public affairs and represent the citizenry of the polis ( in ancient athens the boule was comprised of 500 members ) . the bouleuterion generally was a covered , rectilinear building with stepped seating surrounding a central speaker ’ s well in which an altar was placed . the city of priène has a particularly well-preserved example of this civic structure as does the city of miletus . house greek houses of the archaic and classical periods were relatively simple in design . houses usually were centered on a courtyard that would have been the scene for various ritual activities ; the courtyard also provided natural light for the often small houses . the ground floor rooms would have included kitchen and storage rooms , perhaps an animal pen and a latrine ; the chief room was the andron— site of the male-dominated drinking party ( symposion ) . the quarters for women and children ( gynaikeion ) could be located on the second level ( if present ) and were , in any case , segregated from the mens ’ area . it was not uncommon for houses to be attached to workshops or shops . the houses excavated in the southwest part of the athenian agora had walls of mud brick that rested on stone socles and tiled roofs , with floors of beaten clay . the city of olynthus in chalcidice , greece , destroyed by military action in 348 b.c.e. , preserves many well-appointed courtyard houses arranged within the hippodamian grid-plan of the city . house a vii 4 had a large cobbled courtyard that was used for domestic industry . while some rooms were fairly plain , with earthen floors , the andron was the most well-appointed room of the house . fortifications the mycenaean fortifications of bronze age greece ( c. 1300 b.c.e . ) are particularly well known—the megalithic architecture ( also referred to as cyclopean because of the use of enormous stones ) represents a trend in bronze age architecture . while these massive bronze age walls are difficult to best , first millennium b.c.e . greece also shows evidence for stone built fortification walls . in attika ( the territory of athens ) , a series of classical and hellenistic walls built in ashlar masonry ( squared masonry blocks ) have been studied as a potential system of border defenses . at palairos in epirus ( greece ) the massive fortifications enclose a high citadel that occupies imposing terrain . stadium , gymnasium , and palaestra the greek stadium ( derived from stadion , a greek measurement equivalent to c. 578 feet or 176 meters ) was the location of foot races held as part of sacred games ; these structures are often found in the context of sanctuaries , as in the case of the panhellenic sanctuaries at olympia and epidauros . long and narrow , with a horseshoe shape , the stadium occupied reasonably flat terrain . the gymnasium ( from the greek term gymnós meaning `` naked '' ) was a training center for athletes who participated in public games . this facility tended to include areas for both training and storage . the palaestra ( παλαίστρα ) was an exercise facility originally connected with the training of wrestlers . these complexes were generally rectilinear in plan , with a colonnade framing a central , open space . altar since blood sacrifice was a key component of greek ritual practice , an altar was essential for these purposes . while altars did not necessarily need to be architecturalized , they could be and , in some cases , they assumed a monumental scale . the third century b.c.e . altar of hieron ii at syracuse , sicily , provides one such example . at c. 196 meters in length and c. 11 m in height the massive altar was reported to be capable of hosting the simultaneous sacrifice of 450 bulls ( diodorus siculus history 11.72.2 ) . another spectacular altar is the altar of zeus from pergamon , built during the first half of the second century b.c.e . the altar itself is screened by a monumental enclosure decorated with sculpture ; the monument measures c. 35.64 by 33.4 meters . the altar is best known for its program of relief sculpture that depict a gigantomachy ( battle between the olympian gods and the giants ) that is presented as an allegory for the military conquests of the kings of pergamon . despite its monumental scale and lavish decoration , the pergamon altar preserves the basic and necessary features of the greek altar : it is frontal and approached by stairs and is open to the air—to allow not only for the blood sacrifice itself but also for the burning of the thigh bones and fat as an offering to the gods . fountain house the fountain house is a public building that provides access to clean drinking water and at which water jars and containers could be filled . the southeast fountain house in the athenian agora ( c. 530 b.c.e . ) provides an example of this tendency to position fountain houses and their dependable supply of clean drinking water close to civic spaces like the agora . gathering water was seen as a woman ’ s task and , as such , it offered the often isolated women a chance to socialize with others while collecting water . fountain house scenes are common on ceramic water jars ( hydriai ) , as is the case for a black-figured hydria ( c. 525-500 b.c.e . ) found in an etruscan tomb in vulci that is now in the british museum legacy the architecture of ancient greece influenced ancient roman architecture , and became the architectural vernacular employed in the expansive hellenistic world created in the wake of the conquests of alexander the great . greek architectural forms became implanted so deeply in the roman architectural mindset that they endured throughout antiquity , only to then be re-discovered in the renaissance and especially from the mid-eighteenth century onwards as a feature of the neo-classical movement . this durable legacy helps to explain why the ancient greek architectural orders and the tenets of greek design are still so prevalent—and visible—in our post-modern world . essay by dr. jeffrey a. becker additional resources : athenian agora excavations j. m. camp , the athenian agora : a short guide to the excavations ( american school of classical studies at athens ) architecture in ancient greece on the metropolitan museum of art 's heilbrunn timeline of art history b . a. ault and l. nevett , ancient greek houses and households : chronological , regional , and social diversity ( philadelphia : university of pennsylvania press , 2005 ) . n. cahill , household and city organization at olynthus ( new haven : yale university press , 2001 ) . j. j. coulton , the architectural development of the greek stoa ( oxford : clarendon press , 1976 ) . j. j. coulton , ancient greek architects at work : problems of structure and design ( ithaca ny : cornell university press , 1982 ) . w. b. dinsmoor , the architecture of greece : an account of its historic development 3rd ed . ( london : batsford , 1950 ) . marie-christine hellmann , l ’ architecture grecque 3 vol . ( paris : picard , 2002-2010 ) . m. korres , stones of the parthenon ( los angeles : j. paul getty museum , 2000 ) . a. w. lawrence , greek architecture 5th ed . ( new haven : yale university press , 1996 ) . c. g. malacrino , constructing the ancient world : architectural techniques of the greeks and romans ( los angeles : j. paul getty museum , 2010 ) . a. mazarakis ainian , from rulers ' dwellings to temples : architecture , religion and society in early iron age greece ( 1100-700 b.c . ) ( jonsered : p. åströms förlag , 1997 ) . l. nevett , house and society in the ancient greek world ( cambridge : cambridge university press , 1999 ) . j. ober , fortress attica : defense of the athenian land frontier , 404-322 b.c . ( leiden : e. j. brill , 1985 ) . d. s. robertson , greek and roman architecture 2nd ed . ( cambridge : cambridge university press , 1969 ) . j. n. travlos , pictorial dictionary of ancient athens ( new york : praeger , 1971 ) . f. e. winter , greek fortifications ( toronto : university of toronto press , 1971 ) . f. e. winter , studies in hellenistic architecture ( toronto : university of toronto press , 2006 ) . w. wrede , attische mauern ( athens : deutsches archäologisches institut , 1933 ) . r. e. wycherley , the stones of athens ( princeton : princeton university press , 1978 ) .
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while altars did not necessarily need to be architecturalized , they could be and , in some cases , they assumed a monumental scale . the third century b.c.e . altar of hieron ii at syracuse , sicily , provides one such example .
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in the third paragraph , were there other important types of columns ?
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for most of us , architecture is easy to take for granted . its everywhere in our daily lives—sometimes elegant , other times shabby , but generally ubiquitous . how often do we stop to examine and contemplate its form and style ? stopping for that contemplation offers not only the opportunity to understand one ’ s daily surroundings , but also to appreciate the connection that exists between architectural forms in our own time and those from the past . architectural tradition and design has the ability to link disparate cultures together over time and space—and this is certainly true of the legacy of architectural forms created by the ancient greeks . greek architecture refers to the architecture of the greek-speaking peoples who inhabited the greek mainland and the peloponnese , the islands of the aegean sea , the greek colonies in ionia ( coastal asia minor ) , and magna graecia ( greek colonies in italy and sicily ) . greek architecture stretches from c. 900 b.c.e . to the first century c.e . ( with the earliest extant stone architecture dating to the seventh century b.c.e . ) . greek architecture influenced roman architecture and architects in profound ways , such that roman imperial architecture adopts and incorporates many greek elements into its own practice . an overview of basic building typologies demonstrates the range and diversity of greek architecture . temple the most recognizably “ greek ” structure is the temple ( even though the architecture of greek temples is actually quite diverse ) . the greeks referred to temples with the term ὁ ναός ( ho naós ) meaning `` dwelling ; '' temple derives from the latin term , templum . the earliest shrines were built to honor divinities and were made from materials such as a wood and mud brick—materials that typically do n't survive very long . the basic form of the naos emerges as early as the tenth century b.c.e . as a simple , rectangular room with projecting walls ( antae ) that created a shallow porch . this basic form remained unchanged in its concept for centuries . in the eighth century b.c.e . greek architecture begins to make the move from ephemeral materials ( wood , mud brick , thatch ) to permanent materials ( namely , stone ) . during the archaic period the tenets of the doric order of architecture in the greek mainland became firmly established , leading to a wave of monumental temple building during the sixth and fifth centuries b.c.e . greek city-states invested substantial resources in temple building—as they competed with each other not just in strategic and economic terms , but also in their architecture . for example , athens devoted enormous resources to the construction of the acropolis in the 5th century b.c.e.—in part so that athenians could be confident that the temples built to honor their gods surpassed anything that their rival states could offer . the multi-phase architectural development of sanctuaries such as that of hera on the island of samos demonstrate not only the change that occurred in construction techniques over time but also how the greeks re-used sacred spaces—with the later phases built directly atop the preceding ones . perhaps the fullest , and most famous , expression of classical greek temple architecture is the periclean parthenon of athens—a doric order structure , the parthenon represents the maturity of the greek classical form . greek temples are often categorized in terms of their ground plan and the way in which the columns are arranged . a prostyle temple is a temple that has columns only at the front , while an amphiprostyle temple has columns at the front and the rear . temples with a peripteral arrangement ( from the greek πτερον ( pteron ) meaning `` wing ) have a single line of columns arranged all around the exterior of the temple building . dipteral temples simply have a double row of columns surrounding the building . one of the more unusual plans is the tholos , a temple with a circular ground plan ; famous examples are attested at the sanctuary of apollo in delphi and the sanctuary of asclepius at epidauros . stoa stoa ( στοά ) is a greek architectural term that describes a covered walkway or colonnade that was usually designed for public use . early examples , often employing the doric order , were usually composed of a single level , although later examples ( hellenistic and roman ) came to be two-story freestanding structures . these later examples allowed interior space for shops or other rooms and often incorporated the ionic order for interior colonnades . greek city planners came to prefer the stoa as a device for framing the agora ( public market place ) of a city or town . the south stoa constructed as part of the sanctuary of hera on the island of samos ( c. 700-550 b.c.e . ) numbers among the earliest examples of the stoa in greek architecture . many cities , particularly athens and corinth , came to have elaborate and famous stoas . in athens the famous stoa poikile ( “ painted stoa ” ) , c. fifth century b.c.e. , housed paintings of famous greek military exploits including the battle of marathon , while the stoa basileios ( “ royal stoa ” ) , c. fifth century b.c.e. , was the seat of a chief civic official ( archon basileios ) . later , through the patronage of the kings of pergamon , the athenian agora was augmented by the famed stoa of attalos ( c. 159-138 b.c.e . ) which was recently rebuilt according to the ancient specifications and now houses the archaeological museum for the athenian agora itself ( see image above ) . at corinth the stoa persisted as an architectural type well into the roman period ; the south stoa there ( above ) , c. 150 c.e. , shows the continued utility of this building design for framing civic space . from the hellenistic period onwards the stoa also lent its name to a philosophical school , as zeno of citium ( c. 334-262 b.c.e . ) originally taught his stoic philosophy in the stoa poikile of athens . theater the greek theater was a large , open-air structure used for dramatic performance . theaters often took advantage of hillsides and naturally sloping terrain and , in general , utilized the panoramic landscape as the backdrop to the stage itself . the greek theater is composed of the seating area ( theatron ) , a circular space for the chorus to perform ( orchestra ) , and the stage ( skene ) . tiered seats in the theatron provided space for spectators . two side aisles ( parados , pl . paradoi ) provided access to the orchestra . the greek theater inspired the roman version of the theater directly , although the romans introduced some modifications to the concept of theater architecture . in many cases the romans converted pre-existing greek theaters to conform to their own architectural ideals , as is evident in the theater of dionysos on the slopes of the athenian acropolis . since theatrical performances were often linked to sacred festivals , it is not uncommon to find theaters associated directly with sanctuaries . bouleuterion the bouleuterion ( βουλευτήριον ) was an important civic building in a greek city , as it was the meeting place of the boule ( citizen council ) of the city . these select representatives assembled to handle public affairs and represent the citizenry of the polis ( in ancient athens the boule was comprised of 500 members ) . the bouleuterion generally was a covered , rectilinear building with stepped seating surrounding a central speaker ’ s well in which an altar was placed . the city of priène has a particularly well-preserved example of this civic structure as does the city of miletus . house greek houses of the archaic and classical periods were relatively simple in design . houses usually were centered on a courtyard that would have been the scene for various ritual activities ; the courtyard also provided natural light for the often small houses . the ground floor rooms would have included kitchen and storage rooms , perhaps an animal pen and a latrine ; the chief room was the andron— site of the male-dominated drinking party ( symposion ) . the quarters for women and children ( gynaikeion ) could be located on the second level ( if present ) and were , in any case , segregated from the mens ’ area . it was not uncommon for houses to be attached to workshops or shops . the houses excavated in the southwest part of the athenian agora had walls of mud brick that rested on stone socles and tiled roofs , with floors of beaten clay . the city of olynthus in chalcidice , greece , destroyed by military action in 348 b.c.e. , preserves many well-appointed courtyard houses arranged within the hippodamian grid-plan of the city . house a vii 4 had a large cobbled courtyard that was used for domestic industry . while some rooms were fairly plain , with earthen floors , the andron was the most well-appointed room of the house . fortifications the mycenaean fortifications of bronze age greece ( c. 1300 b.c.e . ) are particularly well known—the megalithic architecture ( also referred to as cyclopean because of the use of enormous stones ) represents a trend in bronze age architecture . while these massive bronze age walls are difficult to best , first millennium b.c.e . greece also shows evidence for stone built fortification walls . in attika ( the territory of athens ) , a series of classical and hellenistic walls built in ashlar masonry ( squared masonry blocks ) have been studied as a potential system of border defenses . at palairos in epirus ( greece ) the massive fortifications enclose a high citadel that occupies imposing terrain . stadium , gymnasium , and palaestra the greek stadium ( derived from stadion , a greek measurement equivalent to c. 578 feet or 176 meters ) was the location of foot races held as part of sacred games ; these structures are often found in the context of sanctuaries , as in the case of the panhellenic sanctuaries at olympia and epidauros . long and narrow , with a horseshoe shape , the stadium occupied reasonably flat terrain . the gymnasium ( from the greek term gymnós meaning `` naked '' ) was a training center for athletes who participated in public games . this facility tended to include areas for both training and storage . the palaestra ( παλαίστρα ) was an exercise facility originally connected with the training of wrestlers . these complexes were generally rectilinear in plan , with a colonnade framing a central , open space . altar since blood sacrifice was a key component of greek ritual practice , an altar was essential for these purposes . while altars did not necessarily need to be architecturalized , they could be and , in some cases , they assumed a monumental scale . the third century b.c.e . altar of hieron ii at syracuse , sicily , provides one such example . at c. 196 meters in length and c. 11 m in height the massive altar was reported to be capable of hosting the simultaneous sacrifice of 450 bulls ( diodorus siculus history 11.72.2 ) . another spectacular altar is the altar of zeus from pergamon , built during the first half of the second century b.c.e . the altar itself is screened by a monumental enclosure decorated with sculpture ; the monument measures c. 35.64 by 33.4 meters . the altar is best known for its program of relief sculpture that depict a gigantomachy ( battle between the olympian gods and the giants ) that is presented as an allegory for the military conquests of the kings of pergamon . despite its monumental scale and lavish decoration , the pergamon altar preserves the basic and necessary features of the greek altar : it is frontal and approached by stairs and is open to the air—to allow not only for the blood sacrifice itself but also for the burning of the thigh bones and fat as an offering to the gods . fountain house the fountain house is a public building that provides access to clean drinking water and at which water jars and containers could be filled . the southeast fountain house in the athenian agora ( c. 530 b.c.e . ) provides an example of this tendency to position fountain houses and their dependable supply of clean drinking water close to civic spaces like the agora . gathering water was seen as a woman ’ s task and , as such , it offered the often isolated women a chance to socialize with others while collecting water . fountain house scenes are common on ceramic water jars ( hydriai ) , as is the case for a black-figured hydria ( c. 525-500 b.c.e . ) found in an etruscan tomb in vulci that is now in the british museum legacy the architecture of ancient greece influenced ancient roman architecture , and became the architectural vernacular employed in the expansive hellenistic world created in the wake of the conquests of alexander the great . greek architectural forms became implanted so deeply in the roman architectural mindset that they endured throughout antiquity , only to then be re-discovered in the renaissance and especially from the mid-eighteenth century onwards as a feature of the neo-classical movement . this durable legacy helps to explain why the ancient greek architectural orders and the tenets of greek design are still so prevalent—and visible—in our post-modern world . essay by dr. jeffrey a. becker additional resources : athenian agora excavations j. m. camp , the athenian agora : a short guide to the excavations ( american school of classical studies at athens ) architecture in ancient greece on the metropolitan museum of art 's heilbrunn timeline of art history b . a. ault and l. nevett , ancient greek houses and households : chronological , regional , and social diversity ( philadelphia : university of pennsylvania press , 2005 ) . n. cahill , household and city organization at olynthus ( new haven : yale university press , 2001 ) . j. j. coulton , the architectural development of the greek stoa ( oxford : clarendon press , 1976 ) . j. j. coulton , ancient greek architects at work : problems of structure and design ( ithaca ny : cornell university press , 1982 ) . w. b. dinsmoor , the architecture of greece : an account of its historic development 3rd ed . ( london : batsford , 1950 ) . marie-christine hellmann , l ’ architecture grecque 3 vol . ( paris : picard , 2002-2010 ) . m. korres , stones of the parthenon ( los angeles : j. paul getty museum , 2000 ) . a. w. lawrence , greek architecture 5th ed . ( new haven : yale university press , 1996 ) . c. g. malacrino , constructing the ancient world : architectural techniques of the greeks and romans ( los angeles : j. paul getty museum , 2010 ) . a. mazarakis ainian , from rulers ' dwellings to temples : architecture , religion and society in early iron age greece ( 1100-700 b.c . ) ( jonsered : p. åströms förlag , 1997 ) . l. nevett , house and society in the ancient greek world ( cambridge : cambridge university press , 1999 ) . j. ober , fortress attica : defense of the athenian land frontier , 404-322 b.c . ( leiden : e. j. brill , 1985 ) . d. s. robertson , greek and roman architecture 2nd ed . ( cambridge : cambridge university press , 1969 ) . j. n. travlos , pictorial dictionary of ancient athens ( new york : praeger , 1971 ) . f. e. winter , greek fortifications ( toronto : university of toronto press , 1971 ) . f. e. winter , studies in hellenistic architecture ( toronto : university of toronto press , 2006 ) . w. wrede , attische mauern ( athens : deutsches archäologisches institut , 1933 ) . r. e. wycherley , the stones of athens ( princeton : princeton university press , 1978 ) .
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originally taught his stoic philosophy in the stoa poikile of athens . theater the greek theater was a large , open-air structure used for dramatic performance . theaters often took advantage of hillsides and naturally sloping terrain and , in general , utilized the panoramic landscape as the backdrop to the stage itself .
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how greek theater is forerunner modern gymnasium ?
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for most of us , architecture is easy to take for granted . its everywhere in our daily lives—sometimes elegant , other times shabby , but generally ubiquitous . how often do we stop to examine and contemplate its form and style ? stopping for that contemplation offers not only the opportunity to understand one ’ s daily surroundings , but also to appreciate the connection that exists between architectural forms in our own time and those from the past . architectural tradition and design has the ability to link disparate cultures together over time and space—and this is certainly true of the legacy of architectural forms created by the ancient greeks . greek architecture refers to the architecture of the greek-speaking peoples who inhabited the greek mainland and the peloponnese , the islands of the aegean sea , the greek colonies in ionia ( coastal asia minor ) , and magna graecia ( greek colonies in italy and sicily ) . greek architecture stretches from c. 900 b.c.e . to the first century c.e . ( with the earliest extant stone architecture dating to the seventh century b.c.e . ) . greek architecture influenced roman architecture and architects in profound ways , such that roman imperial architecture adopts and incorporates many greek elements into its own practice . an overview of basic building typologies demonstrates the range and diversity of greek architecture . temple the most recognizably “ greek ” structure is the temple ( even though the architecture of greek temples is actually quite diverse ) . the greeks referred to temples with the term ὁ ναός ( ho naós ) meaning `` dwelling ; '' temple derives from the latin term , templum . the earliest shrines were built to honor divinities and were made from materials such as a wood and mud brick—materials that typically do n't survive very long . the basic form of the naos emerges as early as the tenth century b.c.e . as a simple , rectangular room with projecting walls ( antae ) that created a shallow porch . this basic form remained unchanged in its concept for centuries . in the eighth century b.c.e . greek architecture begins to make the move from ephemeral materials ( wood , mud brick , thatch ) to permanent materials ( namely , stone ) . during the archaic period the tenets of the doric order of architecture in the greek mainland became firmly established , leading to a wave of monumental temple building during the sixth and fifth centuries b.c.e . greek city-states invested substantial resources in temple building—as they competed with each other not just in strategic and economic terms , but also in their architecture . for example , athens devoted enormous resources to the construction of the acropolis in the 5th century b.c.e.—in part so that athenians could be confident that the temples built to honor their gods surpassed anything that their rival states could offer . the multi-phase architectural development of sanctuaries such as that of hera on the island of samos demonstrate not only the change that occurred in construction techniques over time but also how the greeks re-used sacred spaces—with the later phases built directly atop the preceding ones . perhaps the fullest , and most famous , expression of classical greek temple architecture is the periclean parthenon of athens—a doric order structure , the parthenon represents the maturity of the greek classical form . greek temples are often categorized in terms of their ground plan and the way in which the columns are arranged . a prostyle temple is a temple that has columns only at the front , while an amphiprostyle temple has columns at the front and the rear . temples with a peripteral arrangement ( from the greek πτερον ( pteron ) meaning `` wing ) have a single line of columns arranged all around the exterior of the temple building . dipteral temples simply have a double row of columns surrounding the building . one of the more unusual plans is the tholos , a temple with a circular ground plan ; famous examples are attested at the sanctuary of apollo in delphi and the sanctuary of asclepius at epidauros . stoa stoa ( στοά ) is a greek architectural term that describes a covered walkway or colonnade that was usually designed for public use . early examples , often employing the doric order , were usually composed of a single level , although later examples ( hellenistic and roman ) came to be two-story freestanding structures . these later examples allowed interior space for shops or other rooms and often incorporated the ionic order for interior colonnades . greek city planners came to prefer the stoa as a device for framing the agora ( public market place ) of a city or town . the south stoa constructed as part of the sanctuary of hera on the island of samos ( c. 700-550 b.c.e . ) numbers among the earliest examples of the stoa in greek architecture . many cities , particularly athens and corinth , came to have elaborate and famous stoas . in athens the famous stoa poikile ( “ painted stoa ” ) , c. fifth century b.c.e. , housed paintings of famous greek military exploits including the battle of marathon , while the stoa basileios ( “ royal stoa ” ) , c. fifth century b.c.e. , was the seat of a chief civic official ( archon basileios ) . later , through the patronage of the kings of pergamon , the athenian agora was augmented by the famed stoa of attalos ( c. 159-138 b.c.e . ) which was recently rebuilt according to the ancient specifications and now houses the archaeological museum for the athenian agora itself ( see image above ) . at corinth the stoa persisted as an architectural type well into the roman period ; the south stoa there ( above ) , c. 150 c.e. , shows the continued utility of this building design for framing civic space . from the hellenistic period onwards the stoa also lent its name to a philosophical school , as zeno of citium ( c. 334-262 b.c.e . ) originally taught his stoic philosophy in the stoa poikile of athens . theater the greek theater was a large , open-air structure used for dramatic performance . theaters often took advantage of hillsides and naturally sloping terrain and , in general , utilized the panoramic landscape as the backdrop to the stage itself . the greek theater is composed of the seating area ( theatron ) , a circular space for the chorus to perform ( orchestra ) , and the stage ( skene ) . tiered seats in the theatron provided space for spectators . two side aisles ( parados , pl . paradoi ) provided access to the orchestra . the greek theater inspired the roman version of the theater directly , although the romans introduced some modifications to the concept of theater architecture . in many cases the romans converted pre-existing greek theaters to conform to their own architectural ideals , as is evident in the theater of dionysos on the slopes of the athenian acropolis . since theatrical performances were often linked to sacred festivals , it is not uncommon to find theaters associated directly with sanctuaries . bouleuterion the bouleuterion ( βουλευτήριον ) was an important civic building in a greek city , as it was the meeting place of the boule ( citizen council ) of the city . these select representatives assembled to handle public affairs and represent the citizenry of the polis ( in ancient athens the boule was comprised of 500 members ) . the bouleuterion generally was a covered , rectilinear building with stepped seating surrounding a central speaker ’ s well in which an altar was placed . the city of priène has a particularly well-preserved example of this civic structure as does the city of miletus . house greek houses of the archaic and classical periods were relatively simple in design . houses usually were centered on a courtyard that would have been the scene for various ritual activities ; the courtyard also provided natural light for the often small houses . the ground floor rooms would have included kitchen and storage rooms , perhaps an animal pen and a latrine ; the chief room was the andron— site of the male-dominated drinking party ( symposion ) . the quarters for women and children ( gynaikeion ) could be located on the second level ( if present ) and were , in any case , segregated from the mens ’ area . it was not uncommon for houses to be attached to workshops or shops . the houses excavated in the southwest part of the athenian agora had walls of mud brick that rested on stone socles and tiled roofs , with floors of beaten clay . the city of olynthus in chalcidice , greece , destroyed by military action in 348 b.c.e. , preserves many well-appointed courtyard houses arranged within the hippodamian grid-plan of the city . house a vii 4 had a large cobbled courtyard that was used for domestic industry . while some rooms were fairly plain , with earthen floors , the andron was the most well-appointed room of the house . fortifications the mycenaean fortifications of bronze age greece ( c. 1300 b.c.e . ) are particularly well known—the megalithic architecture ( also referred to as cyclopean because of the use of enormous stones ) represents a trend in bronze age architecture . while these massive bronze age walls are difficult to best , first millennium b.c.e . greece also shows evidence for stone built fortification walls . in attika ( the territory of athens ) , a series of classical and hellenistic walls built in ashlar masonry ( squared masonry blocks ) have been studied as a potential system of border defenses . at palairos in epirus ( greece ) the massive fortifications enclose a high citadel that occupies imposing terrain . stadium , gymnasium , and palaestra the greek stadium ( derived from stadion , a greek measurement equivalent to c. 578 feet or 176 meters ) was the location of foot races held as part of sacred games ; these structures are often found in the context of sanctuaries , as in the case of the panhellenic sanctuaries at olympia and epidauros . long and narrow , with a horseshoe shape , the stadium occupied reasonably flat terrain . the gymnasium ( from the greek term gymnós meaning `` naked '' ) was a training center for athletes who participated in public games . this facility tended to include areas for both training and storage . the palaestra ( παλαίστρα ) was an exercise facility originally connected with the training of wrestlers . these complexes were generally rectilinear in plan , with a colonnade framing a central , open space . altar since blood sacrifice was a key component of greek ritual practice , an altar was essential for these purposes . while altars did not necessarily need to be architecturalized , they could be and , in some cases , they assumed a monumental scale . the third century b.c.e . altar of hieron ii at syracuse , sicily , provides one such example . at c. 196 meters in length and c. 11 m in height the massive altar was reported to be capable of hosting the simultaneous sacrifice of 450 bulls ( diodorus siculus history 11.72.2 ) . another spectacular altar is the altar of zeus from pergamon , built during the first half of the second century b.c.e . the altar itself is screened by a monumental enclosure decorated with sculpture ; the monument measures c. 35.64 by 33.4 meters . the altar is best known for its program of relief sculpture that depict a gigantomachy ( battle between the olympian gods and the giants ) that is presented as an allegory for the military conquests of the kings of pergamon . despite its monumental scale and lavish decoration , the pergamon altar preserves the basic and necessary features of the greek altar : it is frontal and approached by stairs and is open to the air—to allow not only for the blood sacrifice itself but also for the burning of the thigh bones and fat as an offering to the gods . fountain house the fountain house is a public building that provides access to clean drinking water and at which water jars and containers could be filled . the southeast fountain house in the athenian agora ( c. 530 b.c.e . ) provides an example of this tendency to position fountain houses and their dependable supply of clean drinking water close to civic spaces like the agora . gathering water was seen as a woman ’ s task and , as such , it offered the often isolated women a chance to socialize with others while collecting water . fountain house scenes are common on ceramic water jars ( hydriai ) , as is the case for a black-figured hydria ( c. 525-500 b.c.e . ) found in an etruscan tomb in vulci that is now in the british museum legacy the architecture of ancient greece influenced ancient roman architecture , and became the architectural vernacular employed in the expansive hellenistic world created in the wake of the conquests of alexander the great . greek architectural forms became implanted so deeply in the roman architectural mindset that they endured throughout antiquity , only to then be re-discovered in the renaissance and especially from the mid-eighteenth century onwards as a feature of the neo-classical movement . this durable legacy helps to explain why the ancient greek architectural orders and the tenets of greek design are still so prevalent—and visible—in our post-modern world . essay by dr. jeffrey a. becker additional resources : athenian agora excavations j. m. camp , the athenian agora : a short guide to the excavations ( american school of classical studies at athens ) architecture in ancient greece on the metropolitan museum of art 's heilbrunn timeline of art history b . a. ault and l. nevett , ancient greek houses and households : chronological , regional , and social diversity ( philadelphia : university of pennsylvania press , 2005 ) . n. cahill , household and city organization at olynthus ( new haven : yale university press , 2001 ) . j. j. coulton , the architectural development of the greek stoa ( oxford : clarendon press , 1976 ) . j. j. coulton , ancient greek architects at work : problems of structure and design ( ithaca ny : cornell university press , 1982 ) . w. b. dinsmoor , the architecture of greece : an account of its historic development 3rd ed . ( london : batsford , 1950 ) . marie-christine hellmann , l ’ architecture grecque 3 vol . ( paris : picard , 2002-2010 ) . m. korres , stones of the parthenon ( los angeles : j. paul getty museum , 2000 ) . a. w. lawrence , greek architecture 5th ed . ( new haven : yale university press , 1996 ) . c. g. malacrino , constructing the ancient world : architectural techniques of the greeks and romans ( los angeles : j. paul getty museum , 2010 ) . a. mazarakis ainian , from rulers ' dwellings to temples : architecture , religion and society in early iron age greece ( 1100-700 b.c . ) ( jonsered : p. åströms förlag , 1997 ) . l. nevett , house and society in the ancient greek world ( cambridge : cambridge university press , 1999 ) . j. ober , fortress attica : defense of the athenian land frontier , 404-322 b.c . ( leiden : e. j. brill , 1985 ) . d. s. robertson , greek and roman architecture 2nd ed . ( cambridge : cambridge university press , 1969 ) . j. n. travlos , pictorial dictionary of ancient athens ( new york : praeger , 1971 ) . f. e. winter , greek fortifications ( toronto : university of toronto press , 1971 ) . f. e. winter , studies in hellenistic architecture ( toronto : university of toronto press , 2006 ) . w. wrede , attische mauern ( athens : deutsches archäologisches institut , 1933 ) . r. e. wycherley , the stones of athens ( princeton : princeton university press , 1978 ) .
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for most of us , architecture is easy to take for granted . its everywhere in our daily lives—sometimes elegant , other times shabby , but generally ubiquitous . how often do we stop to examine and contemplate its form and style ?
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did the greeks sometimes use the golden numeral to build their structures ?
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for most of us , architecture is easy to take for granted . its everywhere in our daily lives—sometimes elegant , other times shabby , but generally ubiquitous . how often do we stop to examine and contemplate its form and style ? stopping for that contemplation offers not only the opportunity to understand one ’ s daily surroundings , but also to appreciate the connection that exists between architectural forms in our own time and those from the past . architectural tradition and design has the ability to link disparate cultures together over time and space—and this is certainly true of the legacy of architectural forms created by the ancient greeks . greek architecture refers to the architecture of the greek-speaking peoples who inhabited the greek mainland and the peloponnese , the islands of the aegean sea , the greek colonies in ionia ( coastal asia minor ) , and magna graecia ( greek colonies in italy and sicily ) . greek architecture stretches from c. 900 b.c.e . to the first century c.e . ( with the earliest extant stone architecture dating to the seventh century b.c.e . ) . greek architecture influenced roman architecture and architects in profound ways , such that roman imperial architecture adopts and incorporates many greek elements into its own practice . an overview of basic building typologies demonstrates the range and diversity of greek architecture . temple the most recognizably “ greek ” structure is the temple ( even though the architecture of greek temples is actually quite diverse ) . the greeks referred to temples with the term ὁ ναός ( ho naós ) meaning `` dwelling ; '' temple derives from the latin term , templum . the earliest shrines were built to honor divinities and were made from materials such as a wood and mud brick—materials that typically do n't survive very long . the basic form of the naos emerges as early as the tenth century b.c.e . as a simple , rectangular room with projecting walls ( antae ) that created a shallow porch . this basic form remained unchanged in its concept for centuries . in the eighth century b.c.e . greek architecture begins to make the move from ephemeral materials ( wood , mud brick , thatch ) to permanent materials ( namely , stone ) . during the archaic period the tenets of the doric order of architecture in the greek mainland became firmly established , leading to a wave of monumental temple building during the sixth and fifth centuries b.c.e . greek city-states invested substantial resources in temple building—as they competed with each other not just in strategic and economic terms , but also in their architecture . for example , athens devoted enormous resources to the construction of the acropolis in the 5th century b.c.e.—in part so that athenians could be confident that the temples built to honor their gods surpassed anything that their rival states could offer . the multi-phase architectural development of sanctuaries such as that of hera on the island of samos demonstrate not only the change that occurred in construction techniques over time but also how the greeks re-used sacred spaces—with the later phases built directly atop the preceding ones . perhaps the fullest , and most famous , expression of classical greek temple architecture is the periclean parthenon of athens—a doric order structure , the parthenon represents the maturity of the greek classical form . greek temples are often categorized in terms of their ground plan and the way in which the columns are arranged . a prostyle temple is a temple that has columns only at the front , while an amphiprostyle temple has columns at the front and the rear . temples with a peripteral arrangement ( from the greek πτερον ( pteron ) meaning `` wing ) have a single line of columns arranged all around the exterior of the temple building . dipteral temples simply have a double row of columns surrounding the building . one of the more unusual plans is the tholos , a temple with a circular ground plan ; famous examples are attested at the sanctuary of apollo in delphi and the sanctuary of asclepius at epidauros . stoa stoa ( στοά ) is a greek architectural term that describes a covered walkway or colonnade that was usually designed for public use . early examples , often employing the doric order , were usually composed of a single level , although later examples ( hellenistic and roman ) came to be two-story freestanding structures . these later examples allowed interior space for shops or other rooms and often incorporated the ionic order for interior colonnades . greek city planners came to prefer the stoa as a device for framing the agora ( public market place ) of a city or town . the south stoa constructed as part of the sanctuary of hera on the island of samos ( c. 700-550 b.c.e . ) numbers among the earliest examples of the stoa in greek architecture . many cities , particularly athens and corinth , came to have elaborate and famous stoas . in athens the famous stoa poikile ( “ painted stoa ” ) , c. fifth century b.c.e. , housed paintings of famous greek military exploits including the battle of marathon , while the stoa basileios ( “ royal stoa ” ) , c. fifth century b.c.e. , was the seat of a chief civic official ( archon basileios ) . later , through the patronage of the kings of pergamon , the athenian agora was augmented by the famed stoa of attalos ( c. 159-138 b.c.e . ) which was recently rebuilt according to the ancient specifications and now houses the archaeological museum for the athenian agora itself ( see image above ) . at corinth the stoa persisted as an architectural type well into the roman period ; the south stoa there ( above ) , c. 150 c.e. , shows the continued utility of this building design for framing civic space . from the hellenistic period onwards the stoa also lent its name to a philosophical school , as zeno of citium ( c. 334-262 b.c.e . ) originally taught his stoic philosophy in the stoa poikile of athens . theater the greek theater was a large , open-air structure used for dramatic performance . theaters often took advantage of hillsides and naturally sloping terrain and , in general , utilized the panoramic landscape as the backdrop to the stage itself . the greek theater is composed of the seating area ( theatron ) , a circular space for the chorus to perform ( orchestra ) , and the stage ( skene ) . tiered seats in the theatron provided space for spectators . two side aisles ( parados , pl . paradoi ) provided access to the orchestra . the greek theater inspired the roman version of the theater directly , although the romans introduced some modifications to the concept of theater architecture . in many cases the romans converted pre-existing greek theaters to conform to their own architectural ideals , as is evident in the theater of dionysos on the slopes of the athenian acropolis . since theatrical performances were often linked to sacred festivals , it is not uncommon to find theaters associated directly with sanctuaries . bouleuterion the bouleuterion ( βουλευτήριον ) was an important civic building in a greek city , as it was the meeting place of the boule ( citizen council ) of the city . these select representatives assembled to handle public affairs and represent the citizenry of the polis ( in ancient athens the boule was comprised of 500 members ) . the bouleuterion generally was a covered , rectilinear building with stepped seating surrounding a central speaker ’ s well in which an altar was placed . the city of priène has a particularly well-preserved example of this civic structure as does the city of miletus . house greek houses of the archaic and classical periods were relatively simple in design . houses usually were centered on a courtyard that would have been the scene for various ritual activities ; the courtyard also provided natural light for the often small houses . the ground floor rooms would have included kitchen and storage rooms , perhaps an animal pen and a latrine ; the chief room was the andron— site of the male-dominated drinking party ( symposion ) . the quarters for women and children ( gynaikeion ) could be located on the second level ( if present ) and were , in any case , segregated from the mens ’ area . it was not uncommon for houses to be attached to workshops or shops . the houses excavated in the southwest part of the athenian agora had walls of mud brick that rested on stone socles and tiled roofs , with floors of beaten clay . the city of olynthus in chalcidice , greece , destroyed by military action in 348 b.c.e. , preserves many well-appointed courtyard houses arranged within the hippodamian grid-plan of the city . house a vii 4 had a large cobbled courtyard that was used for domestic industry . while some rooms were fairly plain , with earthen floors , the andron was the most well-appointed room of the house . fortifications the mycenaean fortifications of bronze age greece ( c. 1300 b.c.e . ) are particularly well known—the megalithic architecture ( also referred to as cyclopean because of the use of enormous stones ) represents a trend in bronze age architecture . while these massive bronze age walls are difficult to best , first millennium b.c.e . greece also shows evidence for stone built fortification walls . in attika ( the territory of athens ) , a series of classical and hellenistic walls built in ashlar masonry ( squared masonry blocks ) have been studied as a potential system of border defenses . at palairos in epirus ( greece ) the massive fortifications enclose a high citadel that occupies imposing terrain . stadium , gymnasium , and palaestra the greek stadium ( derived from stadion , a greek measurement equivalent to c. 578 feet or 176 meters ) was the location of foot races held as part of sacred games ; these structures are often found in the context of sanctuaries , as in the case of the panhellenic sanctuaries at olympia and epidauros . long and narrow , with a horseshoe shape , the stadium occupied reasonably flat terrain . the gymnasium ( from the greek term gymnós meaning `` naked '' ) was a training center for athletes who participated in public games . this facility tended to include areas for both training and storage . the palaestra ( παλαίστρα ) was an exercise facility originally connected with the training of wrestlers . these complexes were generally rectilinear in plan , with a colonnade framing a central , open space . altar since blood sacrifice was a key component of greek ritual practice , an altar was essential for these purposes . while altars did not necessarily need to be architecturalized , they could be and , in some cases , they assumed a monumental scale . the third century b.c.e . altar of hieron ii at syracuse , sicily , provides one such example . at c. 196 meters in length and c. 11 m in height the massive altar was reported to be capable of hosting the simultaneous sacrifice of 450 bulls ( diodorus siculus history 11.72.2 ) . another spectacular altar is the altar of zeus from pergamon , built during the first half of the second century b.c.e . the altar itself is screened by a monumental enclosure decorated with sculpture ; the monument measures c. 35.64 by 33.4 meters . the altar is best known for its program of relief sculpture that depict a gigantomachy ( battle between the olympian gods and the giants ) that is presented as an allegory for the military conquests of the kings of pergamon . despite its monumental scale and lavish decoration , the pergamon altar preserves the basic and necessary features of the greek altar : it is frontal and approached by stairs and is open to the air—to allow not only for the blood sacrifice itself but also for the burning of the thigh bones and fat as an offering to the gods . fountain house the fountain house is a public building that provides access to clean drinking water and at which water jars and containers could be filled . the southeast fountain house in the athenian agora ( c. 530 b.c.e . ) provides an example of this tendency to position fountain houses and their dependable supply of clean drinking water close to civic spaces like the agora . gathering water was seen as a woman ’ s task and , as such , it offered the often isolated women a chance to socialize with others while collecting water . fountain house scenes are common on ceramic water jars ( hydriai ) , as is the case for a black-figured hydria ( c. 525-500 b.c.e . ) found in an etruscan tomb in vulci that is now in the british museum legacy the architecture of ancient greece influenced ancient roman architecture , and became the architectural vernacular employed in the expansive hellenistic world created in the wake of the conquests of alexander the great . greek architectural forms became implanted so deeply in the roman architectural mindset that they endured throughout antiquity , only to then be re-discovered in the renaissance and especially from the mid-eighteenth century onwards as a feature of the neo-classical movement . this durable legacy helps to explain why the ancient greek architectural orders and the tenets of greek design are still so prevalent—and visible—in our post-modern world . essay by dr. jeffrey a. becker additional resources : athenian agora excavations j. m. camp , the athenian agora : a short guide to the excavations ( american school of classical studies at athens ) architecture in ancient greece on the metropolitan museum of art 's heilbrunn timeline of art history b . a. ault and l. nevett , ancient greek houses and households : chronological , regional , and social diversity ( philadelphia : university of pennsylvania press , 2005 ) . n. cahill , household and city organization at olynthus ( new haven : yale university press , 2001 ) . j. j. coulton , the architectural development of the greek stoa ( oxford : clarendon press , 1976 ) . j. j. coulton , ancient greek architects at work : problems of structure and design ( ithaca ny : cornell university press , 1982 ) . w. b. dinsmoor , the architecture of greece : an account of its historic development 3rd ed . ( london : batsford , 1950 ) . marie-christine hellmann , l ’ architecture grecque 3 vol . ( paris : picard , 2002-2010 ) . m. korres , stones of the parthenon ( los angeles : j. paul getty museum , 2000 ) . a. w. lawrence , greek architecture 5th ed . ( new haven : yale university press , 1996 ) . c. g. malacrino , constructing the ancient world : architectural techniques of the greeks and romans ( los angeles : j. paul getty museum , 2010 ) . a. mazarakis ainian , from rulers ' dwellings to temples : architecture , religion and society in early iron age greece ( 1100-700 b.c . ) ( jonsered : p. åströms förlag , 1997 ) . l. nevett , house and society in the ancient greek world ( cambridge : cambridge university press , 1999 ) . j. ober , fortress attica : defense of the athenian land frontier , 404-322 b.c . ( leiden : e. j. brill , 1985 ) . d. s. robertson , greek and roman architecture 2nd ed . ( cambridge : cambridge university press , 1969 ) . j. n. travlos , pictorial dictionary of ancient athens ( new york : praeger , 1971 ) . f. e. winter , greek fortifications ( toronto : university of toronto press , 1971 ) . f. e. winter , studies in hellenistic architecture ( toronto : university of toronto press , 2006 ) . w. wrede , attische mauern ( athens : deutsches archäologisches institut , 1933 ) . r. e. wycherley , the stones of athens ( princeton : princeton university press , 1978 ) .
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the third century b.c.e . altar of hieron ii at syracuse , sicily , provides one such example . at c. 196 meters in length and c. 11 m in height the massive altar was reported to be capable of hosting the simultaneous sacrifice of 450 bulls ( diodorus siculus history 11.72.2 ) .
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i have found many photographs of the ruins of the altar of hieron ii similar to the one posted in the article , but are there any drawings or models of what it originally looked like ?
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for most of us , architecture is easy to take for granted . its everywhere in our daily lives—sometimes elegant , other times shabby , but generally ubiquitous . how often do we stop to examine and contemplate its form and style ? stopping for that contemplation offers not only the opportunity to understand one ’ s daily surroundings , but also to appreciate the connection that exists between architectural forms in our own time and those from the past . architectural tradition and design has the ability to link disparate cultures together over time and space—and this is certainly true of the legacy of architectural forms created by the ancient greeks . greek architecture refers to the architecture of the greek-speaking peoples who inhabited the greek mainland and the peloponnese , the islands of the aegean sea , the greek colonies in ionia ( coastal asia minor ) , and magna graecia ( greek colonies in italy and sicily ) . greek architecture stretches from c. 900 b.c.e . to the first century c.e . ( with the earliest extant stone architecture dating to the seventh century b.c.e . ) . greek architecture influenced roman architecture and architects in profound ways , such that roman imperial architecture adopts and incorporates many greek elements into its own practice . an overview of basic building typologies demonstrates the range and diversity of greek architecture . temple the most recognizably “ greek ” structure is the temple ( even though the architecture of greek temples is actually quite diverse ) . the greeks referred to temples with the term ὁ ναός ( ho naós ) meaning `` dwelling ; '' temple derives from the latin term , templum . the earliest shrines were built to honor divinities and were made from materials such as a wood and mud brick—materials that typically do n't survive very long . the basic form of the naos emerges as early as the tenth century b.c.e . as a simple , rectangular room with projecting walls ( antae ) that created a shallow porch . this basic form remained unchanged in its concept for centuries . in the eighth century b.c.e . greek architecture begins to make the move from ephemeral materials ( wood , mud brick , thatch ) to permanent materials ( namely , stone ) . during the archaic period the tenets of the doric order of architecture in the greek mainland became firmly established , leading to a wave of monumental temple building during the sixth and fifth centuries b.c.e . greek city-states invested substantial resources in temple building—as they competed with each other not just in strategic and economic terms , but also in their architecture . for example , athens devoted enormous resources to the construction of the acropolis in the 5th century b.c.e.—in part so that athenians could be confident that the temples built to honor their gods surpassed anything that their rival states could offer . the multi-phase architectural development of sanctuaries such as that of hera on the island of samos demonstrate not only the change that occurred in construction techniques over time but also how the greeks re-used sacred spaces—with the later phases built directly atop the preceding ones . perhaps the fullest , and most famous , expression of classical greek temple architecture is the periclean parthenon of athens—a doric order structure , the parthenon represents the maturity of the greek classical form . greek temples are often categorized in terms of their ground plan and the way in which the columns are arranged . a prostyle temple is a temple that has columns only at the front , while an amphiprostyle temple has columns at the front and the rear . temples with a peripteral arrangement ( from the greek πτερον ( pteron ) meaning `` wing ) have a single line of columns arranged all around the exterior of the temple building . dipteral temples simply have a double row of columns surrounding the building . one of the more unusual plans is the tholos , a temple with a circular ground plan ; famous examples are attested at the sanctuary of apollo in delphi and the sanctuary of asclepius at epidauros . stoa stoa ( στοά ) is a greek architectural term that describes a covered walkway or colonnade that was usually designed for public use . early examples , often employing the doric order , were usually composed of a single level , although later examples ( hellenistic and roman ) came to be two-story freestanding structures . these later examples allowed interior space for shops or other rooms and often incorporated the ionic order for interior colonnades . greek city planners came to prefer the stoa as a device for framing the agora ( public market place ) of a city or town . the south stoa constructed as part of the sanctuary of hera on the island of samos ( c. 700-550 b.c.e . ) numbers among the earliest examples of the stoa in greek architecture . many cities , particularly athens and corinth , came to have elaborate and famous stoas . in athens the famous stoa poikile ( “ painted stoa ” ) , c. fifth century b.c.e. , housed paintings of famous greek military exploits including the battle of marathon , while the stoa basileios ( “ royal stoa ” ) , c. fifth century b.c.e. , was the seat of a chief civic official ( archon basileios ) . later , through the patronage of the kings of pergamon , the athenian agora was augmented by the famed stoa of attalos ( c. 159-138 b.c.e . ) which was recently rebuilt according to the ancient specifications and now houses the archaeological museum for the athenian agora itself ( see image above ) . at corinth the stoa persisted as an architectural type well into the roman period ; the south stoa there ( above ) , c. 150 c.e. , shows the continued utility of this building design for framing civic space . from the hellenistic period onwards the stoa also lent its name to a philosophical school , as zeno of citium ( c. 334-262 b.c.e . ) originally taught his stoic philosophy in the stoa poikile of athens . theater the greek theater was a large , open-air structure used for dramatic performance . theaters often took advantage of hillsides and naturally sloping terrain and , in general , utilized the panoramic landscape as the backdrop to the stage itself . the greek theater is composed of the seating area ( theatron ) , a circular space for the chorus to perform ( orchestra ) , and the stage ( skene ) . tiered seats in the theatron provided space for spectators . two side aisles ( parados , pl . paradoi ) provided access to the orchestra . the greek theater inspired the roman version of the theater directly , although the romans introduced some modifications to the concept of theater architecture . in many cases the romans converted pre-existing greek theaters to conform to their own architectural ideals , as is evident in the theater of dionysos on the slopes of the athenian acropolis . since theatrical performances were often linked to sacred festivals , it is not uncommon to find theaters associated directly with sanctuaries . bouleuterion the bouleuterion ( βουλευτήριον ) was an important civic building in a greek city , as it was the meeting place of the boule ( citizen council ) of the city . these select representatives assembled to handle public affairs and represent the citizenry of the polis ( in ancient athens the boule was comprised of 500 members ) . the bouleuterion generally was a covered , rectilinear building with stepped seating surrounding a central speaker ’ s well in which an altar was placed . the city of priène has a particularly well-preserved example of this civic structure as does the city of miletus . house greek houses of the archaic and classical periods were relatively simple in design . houses usually were centered on a courtyard that would have been the scene for various ritual activities ; the courtyard also provided natural light for the often small houses . the ground floor rooms would have included kitchen and storage rooms , perhaps an animal pen and a latrine ; the chief room was the andron— site of the male-dominated drinking party ( symposion ) . the quarters for women and children ( gynaikeion ) could be located on the second level ( if present ) and were , in any case , segregated from the mens ’ area . it was not uncommon for houses to be attached to workshops or shops . the houses excavated in the southwest part of the athenian agora had walls of mud brick that rested on stone socles and tiled roofs , with floors of beaten clay . the city of olynthus in chalcidice , greece , destroyed by military action in 348 b.c.e. , preserves many well-appointed courtyard houses arranged within the hippodamian grid-plan of the city . house a vii 4 had a large cobbled courtyard that was used for domestic industry . while some rooms were fairly plain , with earthen floors , the andron was the most well-appointed room of the house . fortifications the mycenaean fortifications of bronze age greece ( c. 1300 b.c.e . ) are particularly well known—the megalithic architecture ( also referred to as cyclopean because of the use of enormous stones ) represents a trend in bronze age architecture . while these massive bronze age walls are difficult to best , first millennium b.c.e . greece also shows evidence for stone built fortification walls . in attika ( the territory of athens ) , a series of classical and hellenistic walls built in ashlar masonry ( squared masonry blocks ) have been studied as a potential system of border defenses . at palairos in epirus ( greece ) the massive fortifications enclose a high citadel that occupies imposing terrain . stadium , gymnasium , and palaestra the greek stadium ( derived from stadion , a greek measurement equivalent to c. 578 feet or 176 meters ) was the location of foot races held as part of sacred games ; these structures are often found in the context of sanctuaries , as in the case of the panhellenic sanctuaries at olympia and epidauros . long and narrow , with a horseshoe shape , the stadium occupied reasonably flat terrain . the gymnasium ( from the greek term gymnós meaning `` naked '' ) was a training center for athletes who participated in public games . this facility tended to include areas for both training and storage . the palaestra ( παλαίστρα ) was an exercise facility originally connected with the training of wrestlers . these complexes were generally rectilinear in plan , with a colonnade framing a central , open space . altar since blood sacrifice was a key component of greek ritual practice , an altar was essential for these purposes . while altars did not necessarily need to be architecturalized , they could be and , in some cases , they assumed a monumental scale . the third century b.c.e . altar of hieron ii at syracuse , sicily , provides one such example . at c. 196 meters in length and c. 11 m in height the massive altar was reported to be capable of hosting the simultaneous sacrifice of 450 bulls ( diodorus siculus history 11.72.2 ) . another spectacular altar is the altar of zeus from pergamon , built during the first half of the second century b.c.e . the altar itself is screened by a monumental enclosure decorated with sculpture ; the monument measures c. 35.64 by 33.4 meters . the altar is best known for its program of relief sculpture that depict a gigantomachy ( battle between the olympian gods and the giants ) that is presented as an allegory for the military conquests of the kings of pergamon . despite its monumental scale and lavish decoration , the pergamon altar preserves the basic and necessary features of the greek altar : it is frontal and approached by stairs and is open to the air—to allow not only for the blood sacrifice itself but also for the burning of the thigh bones and fat as an offering to the gods . fountain house the fountain house is a public building that provides access to clean drinking water and at which water jars and containers could be filled . the southeast fountain house in the athenian agora ( c. 530 b.c.e . ) provides an example of this tendency to position fountain houses and their dependable supply of clean drinking water close to civic spaces like the agora . gathering water was seen as a woman ’ s task and , as such , it offered the often isolated women a chance to socialize with others while collecting water . fountain house scenes are common on ceramic water jars ( hydriai ) , as is the case for a black-figured hydria ( c. 525-500 b.c.e . ) found in an etruscan tomb in vulci that is now in the british museum legacy the architecture of ancient greece influenced ancient roman architecture , and became the architectural vernacular employed in the expansive hellenistic world created in the wake of the conquests of alexander the great . greek architectural forms became implanted so deeply in the roman architectural mindset that they endured throughout antiquity , only to then be re-discovered in the renaissance and especially from the mid-eighteenth century onwards as a feature of the neo-classical movement . this durable legacy helps to explain why the ancient greek architectural orders and the tenets of greek design are still so prevalent—and visible—in our post-modern world . essay by dr. jeffrey a. becker additional resources : athenian agora excavations j. m. camp , the athenian agora : a short guide to the excavations ( american school of classical studies at athens ) architecture in ancient greece on the metropolitan museum of art 's heilbrunn timeline of art history b . a. ault and l. nevett , ancient greek houses and households : chronological , regional , and social diversity ( philadelphia : university of pennsylvania press , 2005 ) . n. cahill , household and city organization at olynthus ( new haven : yale university press , 2001 ) . j. j. coulton , the architectural development of the greek stoa ( oxford : clarendon press , 1976 ) . j. j. coulton , ancient greek architects at work : problems of structure and design ( ithaca ny : cornell university press , 1982 ) . w. b. dinsmoor , the architecture of greece : an account of its historic development 3rd ed . ( london : batsford , 1950 ) . marie-christine hellmann , l ’ architecture grecque 3 vol . ( paris : picard , 2002-2010 ) . m. korres , stones of the parthenon ( los angeles : j. paul getty museum , 2000 ) . a. w. lawrence , greek architecture 5th ed . ( new haven : yale university press , 1996 ) . c. g. malacrino , constructing the ancient world : architectural techniques of the greeks and romans ( los angeles : j. paul getty museum , 2010 ) . a. mazarakis ainian , from rulers ' dwellings to temples : architecture , religion and society in early iron age greece ( 1100-700 b.c . ) ( jonsered : p. åströms förlag , 1997 ) . l. nevett , house and society in the ancient greek world ( cambridge : cambridge university press , 1999 ) . j. ober , fortress attica : defense of the athenian land frontier , 404-322 b.c . ( leiden : e. j. brill , 1985 ) . d. s. robertson , greek and roman architecture 2nd ed . ( cambridge : cambridge university press , 1969 ) . j. n. travlos , pictorial dictionary of ancient athens ( new york : praeger , 1971 ) . f. e. winter , greek fortifications ( toronto : university of toronto press , 1971 ) . f. e. winter , studies in hellenistic architecture ( toronto : university of toronto press , 2006 ) . w. wrede , attische mauern ( athens : deutsches archäologisches institut , 1933 ) . r. e. wycherley , the stones of athens ( princeton : princeton university press , 1978 ) .
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( with the earliest extant stone architecture dating to the seventh century b.c.e . ) . greek architecture influenced roman architecture and architects in profound ways , such that roman imperial architecture adopts and incorporates many greek elements into its own practice . an overview of basic building typologies demonstrates the range and diversity of greek architecture . temple the most recognizably “ greek ” structure is the temple ( even though the architecture of greek temples is actually quite diverse ) .
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how does greek architecture influence modern architecture ?
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coins have a unique significance in the history of aksum . they are particularly important because they provide evidence of aksum and its rulers . the inscriptions on the coins highlight the fact that aksumites were a literate people with knowledge of both ethiopic and greek languages . conversion to christianity it is generally thought that the first aksumite coins were intended for international trade . these coins , bearing the name of king endybis ( c. 270/290 c.e . ) , were mainly struck in gold and silver and followed the weight standard which existed in the roman empire . initially , the symbols of the crescent and disc , which were common to the religions in south arabia to which aksum adhered , were used on early aksumite coins . however , after the conversion of king ezana around 340–356 c.e. , a powerful statement was conveyed by the king ’ s decision to replace the existing symbols with a cross which clearly denoted the importance that christianity had in the kingdom . the coins also had a portrait of the ruler on the obverse and reverse of the coin along with teff , a local type of wheat . inscriptions were another form of information included on the coins . for the most part , gold coins were inscribed in greek and often intended for exports , while silver and copper coins were insribed in ge ’ ez ( aksumite script ) . from the 4th century c.e. , an increasing number of copper coins were issued which had evidently christian inscriptions such as `` joy and peace to the people '' and `` he conquers through christ . '' with the replacement of gold coins with copper ones , the craftsmen of aksum started using specialized techniques of gilding , which was unique to the kingdom and involved gold leaf being added to crowns and other symbols to enhance the appearance , and most probably the value , of coins . this coin was minted during the reign of king joel in sixth-century aksum . between the second and the ninth centuries , the kingdom of aksum prospered in ethiopia . the trade routes along the nile valley that led to the red sea and on into the indian ocean made aksum a destination for many merchants and travelers . the large cross on the reverse of the coin symbolizes the country 's shift to christianity . this took place during the fourth century when a traveller named frumentius converted aksum 's ruler , king ezana . the old religious symbols of the sun and the moon no longer appeared on coins and were replaced with a cross , which was enlarged over the years . the religious symbolism on these coins had strong political implications , as it aligned aksum 's religious identity with its main trading partners , rome and later byzantium . suggested readings : j. williams ( ed . ) , money : a history ( london , the british museum press , 1997 ) . © trustees of the british museum
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they are particularly important because they provide evidence of aksum and its rulers . the inscriptions on the coins highlight the fact that aksumites were a literate people with knowledge of both ethiopic and greek languages . conversion to christianity it is generally thought that the first aksumite coins were intended for international trade .
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were there people who carved the coins as their jobs ?
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coins have a unique significance in the history of aksum . they are particularly important because they provide evidence of aksum and its rulers . the inscriptions on the coins highlight the fact that aksumites were a literate people with knowledge of both ethiopic and greek languages . conversion to christianity it is generally thought that the first aksumite coins were intended for international trade . these coins , bearing the name of king endybis ( c. 270/290 c.e . ) , were mainly struck in gold and silver and followed the weight standard which existed in the roman empire . initially , the symbols of the crescent and disc , which were common to the religions in south arabia to which aksum adhered , were used on early aksumite coins . however , after the conversion of king ezana around 340–356 c.e. , a powerful statement was conveyed by the king ’ s decision to replace the existing symbols with a cross which clearly denoted the importance that christianity had in the kingdom . the coins also had a portrait of the ruler on the obverse and reverse of the coin along with teff , a local type of wheat . inscriptions were another form of information included on the coins . for the most part , gold coins were inscribed in greek and often intended for exports , while silver and copper coins were insribed in ge ’ ez ( aksumite script ) . from the 4th century c.e. , an increasing number of copper coins were issued which had evidently christian inscriptions such as `` joy and peace to the people '' and `` he conquers through christ . '' with the replacement of gold coins with copper ones , the craftsmen of aksum started using specialized techniques of gilding , which was unique to the kingdom and involved gold leaf being added to crowns and other symbols to enhance the appearance , and most probably the value , of coins . this coin was minted during the reign of king joel in sixth-century aksum . between the second and the ninth centuries , the kingdom of aksum prospered in ethiopia . the trade routes along the nile valley that led to the red sea and on into the indian ocean made aksum a destination for many merchants and travelers . the large cross on the reverse of the coin symbolizes the country 's shift to christianity . this took place during the fourth century when a traveller named frumentius converted aksum 's ruler , king ezana . the old religious symbols of the sun and the moon no longer appeared on coins and were replaced with a cross , which was enlarged over the years . the religious symbolism on these coins had strong political implications , as it aligned aksum 's religious identity with its main trading partners , rome and later byzantium . suggested readings : j. williams ( ed . ) , money : a history ( london , the british museum press , 1997 ) . © trustees of the british museum
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the old religious symbols of the sun and the moon no longer appeared on coins and were replaced with a cross , which was enlarged over the years . the religious symbolism on these coins had strong political implications , as it aligned aksum 's religious identity with its main trading partners , rome and later byzantium . suggested readings : j. williams ( ed .
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i mean like , if you carve coins based on religious events , would n't people who did n't believe in the religion get offended at all ?
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coins have a unique significance in the history of aksum . they are particularly important because they provide evidence of aksum and its rulers . the inscriptions on the coins highlight the fact that aksumites were a literate people with knowledge of both ethiopic and greek languages . conversion to christianity it is generally thought that the first aksumite coins were intended for international trade . these coins , bearing the name of king endybis ( c. 270/290 c.e . ) , were mainly struck in gold and silver and followed the weight standard which existed in the roman empire . initially , the symbols of the crescent and disc , which were common to the religions in south arabia to which aksum adhered , were used on early aksumite coins . however , after the conversion of king ezana around 340–356 c.e. , a powerful statement was conveyed by the king ’ s decision to replace the existing symbols with a cross which clearly denoted the importance that christianity had in the kingdom . the coins also had a portrait of the ruler on the obverse and reverse of the coin along with teff , a local type of wheat . inscriptions were another form of information included on the coins . for the most part , gold coins were inscribed in greek and often intended for exports , while silver and copper coins were insribed in ge ’ ez ( aksumite script ) . from the 4th century c.e. , an increasing number of copper coins were issued which had evidently christian inscriptions such as `` joy and peace to the people '' and `` he conquers through christ . '' with the replacement of gold coins with copper ones , the craftsmen of aksum started using specialized techniques of gilding , which was unique to the kingdom and involved gold leaf being added to crowns and other symbols to enhance the appearance , and most probably the value , of coins . this coin was minted during the reign of king joel in sixth-century aksum . between the second and the ninth centuries , the kingdom of aksum prospered in ethiopia . the trade routes along the nile valley that led to the red sea and on into the indian ocean made aksum a destination for many merchants and travelers . the large cross on the reverse of the coin symbolizes the country 's shift to christianity . this took place during the fourth century when a traveller named frumentius converted aksum 's ruler , king ezana . the old religious symbols of the sun and the moon no longer appeared on coins and were replaced with a cross , which was enlarged over the years . the religious symbolism on these coins had strong political implications , as it aligned aksum 's religious identity with its main trading partners , rome and later byzantium . suggested readings : j. williams ( ed . ) , money : a history ( london , the british museum press , 1997 ) . © trustees of the british museum
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the coins also had a portrait of the ruler on the obverse and reverse of the coin along with teff , a local type of wheat . inscriptions were another form of information included on the coins . for the most part , gold coins were inscribed in greek and often intended for exports , while silver and copper coins were insribed in ge ’ ez ( aksumite script ) . from the 4th century c.e. , an increasing number of copper coins were issued which had evidently christian inscriptions such as `` joy and peace to the people '' and `` he conquers through christ . ''
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like , what if people who made the coins were a different religion , and they decided that they were gon na make their own coins the entire time , would n't they get punished and neglected ?
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coins have a unique significance in the history of aksum . they are particularly important because they provide evidence of aksum and its rulers . the inscriptions on the coins highlight the fact that aksumites were a literate people with knowledge of both ethiopic and greek languages . conversion to christianity it is generally thought that the first aksumite coins were intended for international trade . these coins , bearing the name of king endybis ( c. 270/290 c.e . ) , were mainly struck in gold and silver and followed the weight standard which existed in the roman empire . initially , the symbols of the crescent and disc , which were common to the religions in south arabia to which aksum adhered , were used on early aksumite coins . however , after the conversion of king ezana around 340–356 c.e. , a powerful statement was conveyed by the king ’ s decision to replace the existing symbols with a cross which clearly denoted the importance that christianity had in the kingdom . the coins also had a portrait of the ruler on the obverse and reverse of the coin along with teff , a local type of wheat . inscriptions were another form of information included on the coins . for the most part , gold coins were inscribed in greek and often intended for exports , while silver and copper coins were insribed in ge ’ ez ( aksumite script ) . from the 4th century c.e. , an increasing number of copper coins were issued which had evidently christian inscriptions such as `` joy and peace to the people '' and `` he conquers through christ . '' with the replacement of gold coins with copper ones , the craftsmen of aksum started using specialized techniques of gilding , which was unique to the kingdom and involved gold leaf being added to crowns and other symbols to enhance the appearance , and most probably the value , of coins . this coin was minted during the reign of king joel in sixth-century aksum . between the second and the ninth centuries , the kingdom of aksum prospered in ethiopia . the trade routes along the nile valley that led to the red sea and on into the indian ocean made aksum a destination for many merchants and travelers . the large cross on the reverse of the coin symbolizes the country 's shift to christianity . this took place during the fourth century when a traveller named frumentius converted aksum 's ruler , king ezana . the old religious symbols of the sun and the moon no longer appeared on coins and were replaced with a cross , which was enlarged over the years . the religious symbolism on these coins had strong political implications , as it aligned aksum 's religious identity with its main trading partners , rome and later byzantium . suggested readings : j. williams ( ed . ) , money : a history ( london , the british museum press , 1997 ) . © trustees of the british museum
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they are particularly important because they provide evidence of aksum and its rulers . the inscriptions on the coins highlight the fact that aksumites were a literate people with knowledge of both ethiopic and greek languages . conversion to christianity it is generally thought that the first aksumite coins were intended for international trade .
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were there people who carved the coins as their jobs ?
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maybe you can bring it with you…if you are rich enough . the elite men and women of the han dynasty ( china's second imperial dynasty , 206 b.c.e.–220 c.e . ) enjoyed an opulent lifestyle that could stretch into the afterlife . today , the well-furnished tombs of the elite give us a glimpse of the luxurious goods they treasured and enjoyed . for instance , a wealthy official could afford beautiful silk robes in contrast to the homespun or paper garments of a laborer or peasant . their tombs also inform us about their cosmological beliefs . marquis of dai , lady dai , and a son three elite tombs , discovered in 1972 , at mawangdui , hunan province ( eastern china ) rank amongst the greatest archeological discoveries in china during the 20th century . they are the tombs of a high-ranking han official civil servant , the marquis of dai , lady dai ( his wife ) , and their son . the marquis died in 186 b.c.e. , and his wife and son both died by 163 b.c.e . the marquis ’ tomb was not in good condition when it was discovered . however , the objects in the son 's and wife 's tombs were of extraordinary quality and very well preserved . from these objects , we can see that lady dai and her son were to spend the afterlife in sumptuous comfort . in lady dai 's tomb , archaeologists found a painted silk banner over six feet long in excellent condition . the t-shaped banner was on top of the innermost of four nesting coffins . although scholars still debate the function of these banners , we know they had some connection with the afterlife . they may be “ name banners ” used to identify the dead during the mourning ceremonies , or they may have been burial shrouds intended to aid the soul in its passage to the afterlife . lady dai ’ s banner is important for two primary reasons . it is an early example of pictorial ( representing naturalistic scenes not just abstract shapes ) art in china . secondly , the banner features the earliest known portrait in chinese painting . we can divide lady dai ’ s banner into four horizontal registers ( see diagram ) . in the lower central register , we see lady dai in an embroidered silk robe leaning on a staff . this remarkable portrait of lady dai is the earliest example of a painted portrait of a specific individual in china . she stands on a platform along with her servants–two in front and three behind . long , sinuous dragons frame the scene on either side , and their white and pink bodies loop through a bi ( a disc with a hole thought to represent the sky ) underneath lady dai . we understand that this is not a portrait of lady dai in her former life , but an image of her in the afterlife enjoying the immortal comforts of her tomb as she ascends toward the heavens . in the register below the scene of lady dai , we see sacrificial funerary rituals taking place in a mourning hall . tripod containers and vase-shaped vessels for offering food and wine stand in the foreground . in the middle ground , seated mourners line up in two rows . look for the mound in the center , between the two rows of mourners . if you look closely , you can see the patterns on the silk that match the robe lady dai wears in the scene above . her corpse is wrapped in her finest robe ! more vessels appear on a shelf in the background . in the mourning scene , we can also appreciate the importance of lady dai ’ s banner for understanding how artists began to represent depth and space in early chinese painting . they made efforts to indicate depth through the use of the overlapping bodies of the mourners . they also made objects in the foreground larger , and objects in the background smaller , to create the illusion of space in the mourning hall . the afterlife in han dynasty china lady dai ’ s banner gives us some insight into cosmological beliefs and funeral practices of han dynasty china . above and below the scenes of lady dai and the mourning hall , we see images of heaven and the underworld . toward the top , near the cross of the “ t , ” two men face each other and guard the gate to the heavenly realm . directly above the two men , at the very top of the banner , we see a deity with a human head and a dragon body . on the left , a toad standing on a crescent moon flanks the dragon/human deity . on the right , we see what may be a three-legged crow within a pink sun . the moon and the sun are emblematic of a supernatural realm above the human world . dragons and other immortal beings populate the sky . in the lower register , beneath the mourning hall , we see the underworld populated by two giant black fish , a red snake , a pair of blue goats , and an unidentified earthly deity . the deity appears to hold up the floor of the mourning hall , while the two fish cross to form a circle beneath him . the beings in the underworld symbolize water and earth , and they indicate an underground domain below the human world . four compartments surrounded lady dai ’ s central tomb , and they offer some sense of the life she was expected to lead in the afterlife . the top compartment represented a room where lady dai was supposed to sit while having her meal . in this compartment , researchers found cushions , an armrest and her walking stick . the compartment also contained a meal laid out for her to eat in the afterlife . lady dai was 50 years old when she died , but her lavish tomb—marked by her funeral banner —ensured that she would enjoy the comforts of her earthly life for eternity . essay by dr. jennifer n. mcintire , with the assistance of dr. wu hung additional resources : this object at the han provincial museum
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the moon and the sun are emblematic of a supernatural realm above the human world . dragons and other immortal beings populate the sky . in the lower register , beneath the mourning hall , we see the underworld populated by two giant black fish , a red snake , a pair of blue goats , and an unidentified earthly deity .
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why are dragons used , what is the symbolism ?
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maybe you can bring it with you…if you are rich enough . the elite men and women of the han dynasty ( china's second imperial dynasty , 206 b.c.e.–220 c.e . ) enjoyed an opulent lifestyle that could stretch into the afterlife . today , the well-furnished tombs of the elite give us a glimpse of the luxurious goods they treasured and enjoyed . for instance , a wealthy official could afford beautiful silk robes in contrast to the homespun or paper garments of a laborer or peasant . their tombs also inform us about their cosmological beliefs . marquis of dai , lady dai , and a son three elite tombs , discovered in 1972 , at mawangdui , hunan province ( eastern china ) rank amongst the greatest archeological discoveries in china during the 20th century . they are the tombs of a high-ranking han official civil servant , the marquis of dai , lady dai ( his wife ) , and their son . the marquis died in 186 b.c.e. , and his wife and son both died by 163 b.c.e . the marquis ’ tomb was not in good condition when it was discovered . however , the objects in the son 's and wife 's tombs were of extraordinary quality and very well preserved . from these objects , we can see that lady dai and her son were to spend the afterlife in sumptuous comfort . in lady dai 's tomb , archaeologists found a painted silk banner over six feet long in excellent condition . the t-shaped banner was on top of the innermost of four nesting coffins . although scholars still debate the function of these banners , we know they had some connection with the afterlife . they may be “ name banners ” used to identify the dead during the mourning ceremonies , or they may have been burial shrouds intended to aid the soul in its passage to the afterlife . lady dai ’ s banner is important for two primary reasons . it is an early example of pictorial ( representing naturalistic scenes not just abstract shapes ) art in china . secondly , the banner features the earliest known portrait in chinese painting . we can divide lady dai ’ s banner into four horizontal registers ( see diagram ) . in the lower central register , we see lady dai in an embroidered silk robe leaning on a staff . this remarkable portrait of lady dai is the earliest example of a painted portrait of a specific individual in china . she stands on a platform along with her servants–two in front and three behind . long , sinuous dragons frame the scene on either side , and their white and pink bodies loop through a bi ( a disc with a hole thought to represent the sky ) underneath lady dai . we understand that this is not a portrait of lady dai in her former life , but an image of her in the afterlife enjoying the immortal comforts of her tomb as she ascends toward the heavens . in the register below the scene of lady dai , we see sacrificial funerary rituals taking place in a mourning hall . tripod containers and vase-shaped vessels for offering food and wine stand in the foreground . in the middle ground , seated mourners line up in two rows . look for the mound in the center , between the two rows of mourners . if you look closely , you can see the patterns on the silk that match the robe lady dai wears in the scene above . her corpse is wrapped in her finest robe ! more vessels appear on a shelf in the background . in the mourning scene , we can also appreciate the importance of lady dai ’ s banner for understanding how artists began to represent depth and space in early chinese painting . they made efforts to indicate depth through the use of the overlapping bodies of the mourners . they also made objects in the foreground larger , and objects in the background smaller , to create the illusion of space in the mourning hall . the afterlife in han dynasty china lady dai ’ s banner gives us some insight into cosmological beliefs and funeral practices of han dynasty china . above and below the scenes of lady dai and the mourning hall , we see images of heaven and the underworld . toward the top , near the cross of the “ t , ” two men face each other and guard the gate to the heavenly realm . directly above the two men , at the very top of the banner , we see a deity with a human head and a dragon body . on the left , a toad standing on a crescent moon flanks the dragon/human deity . on the right , we see what may be a three-legged crow within a pink sun . the moon and the sun are emblematic of a supernatural realm above the human world . dragons and other immortal beings populate the sky . in the lower register , beneath the mourning hall , we see the underworld populated by two giant black fish , a red snake , a pair of blue goats , and an unidentified earthly deity . the deity appears to hold up the floor of the mourning hall , while the two fish cross to form a circle beneath him . the beings in the underworld symbolize water and earth , and they indicate an underground domain below the human world . four compartments surrounded lady dai ’ s central tomb , and they offer some sense of the life she was expected to lead in the afterlife . the top compartment represented a room where lady dai was supposed to sit while having her meal . in this compartment , researchers found cushions , an armrest and her walking stick . the compartment also contained a meal laid out for her to eat in the afterlife . lady dai was 50 years old when she died , but her lavish tomb—marked by her funeral banner —ensured that she would enjoy the comforts of her earthly life for eternity . essay by dr. jennifer n. mcintire , with the assistance of dr. wu hung additional resources : this object at the han provincial museum
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today , the well-furnished tombs of the elite give us a glimpse of the luxurious goods they treasured and enjoyed . for instance , a wealthy official could afford beautiful silk robes in contrast to the homespun or paper garments of a laborer or peasant . their tombs also inform us about their cosmological beliefs .
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did i read correctly that a chinese peasant might have worn `` paper '' garments ?
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`` discovery '' between 1933 and 1940 , camel corps officer lieutenant brenans of the french foreign legion completed a series of small sketches and hand written notes detailing his discovery of dozens of rock art sites deep within the canyons of the tassili n ’ ajjer . tassili n ’ ajjer is a difficult to access plateau in the algerian section of the sahara desert near the borders of libya and niger in northern africa ( see map below ) . brenans donated hundreds of his sketches to the bardo museum in algiers , alerting the scientific community to one of the richest rock art concentrations on earth and prompting site visits that included fellow frenchman and archaeologist henri lhote . lhote recognized the importance of the region and returned again and again , most notably in 1956 with a team of copyists for a 16 month expedition to map and study the rock art of the tassili . two years later lhote published a la découverte des fresques du tassili . the book became an instant best-seller , and today is one of the most popular texts on archaeological discovery . lhote made african rock art famous by bringing some of the estimated 15,000 human figure and animal paintings and engravings found on the rock walls of the tassili ’ s many gorges and shelters it to the wider public . yet contrary to the impression left by the title of his book , neither lhote nor his team could lay claim to having discovered central saharan rock art : long before lhote , and even before brenans , in the late nineteenth century a number of travelers from germany , switzerland , and france had noted the existence of “ strange ” and “ important ” rock sculptures in ghat , tadrart acacus , and upper tassili . but it was the tuareg—the indigenous peoples of the region , many of whom served as guides to these early european explorers—who long knew of the paintings and engravings covering the rock faces of the tassili . tassili n ’ ajjer is a tamahaq name meaning “ plateau ” of the ajjer people ( the kel ajjer is group of tribes whose traditional territory was here ) . much of the 1,500-2,100 meter high plateau is protected by an 80,000 square kilometer national park . the `` horned goddess '' lhote published not only reproductions of the paintings and engravings he found on the rock walls of the tassili , but also his observations . in one excerpt he reported that with a can of water and a sponge in hand he set out to investigate a “ curious figure ” spotted by a member of his team in an isolated rock shelter located within a compact group of mountains known as the aouanrhet massif , the highest of all the “ rock cities ” on the tassili . lhote swabbed the wall with water to reveal a figure he called the `` horned goddess '' : on the damp rock surface stood out the gracious silhouette of a woman running . one of her legs , slightly flexed , just touched the ground , while the other was raised in the air as high as it would normally go . from the knees , the belt and the widely outstretched arms fell fine fringes . from either side of the head and above two horns that spread out horizontally was an extensive dotted area resembling a cloud of grain falling from a wheat field . although the whole assemblage was skillfully and carefully composed there was something free and easy about it… the running horned woman , the title by which the painting is commonly known today , was found in a massif so secluded and so difficult to access that lhote ’ s team concluded that the collection of shelters was likely a sanctuary and the female figure— “ the most beautiful , the most finished and the most original ” —a goddess : `` perhaps we have here the figure of a priestess of some agricultural religion or the picture of a goddess of such a cult who foreshadow—or is derived from—the goddess isis , to whom , in egypt , was attributed the discovery of agriculture . '' lhote ’ s suggestion that the painting ’ s source was egyptian was influenced by a recently published hypothesis by his mentor , the french anthropologist henri breuil , the then undisputed authority on prehistoric rock art who was renowned for his work on paleolithic cave art in europe . in an essay titled , `` the white lady of brandberg , south-west africa , her companions and her guards , '' breuil famously claimed that a painting discovered in a small rock shelter in namibia showed influences of classical antiquity and was not african in origin , but possibly the work of phoenician travellers from the mediterranean . lhote , equally convinced of outside influence , linked the tassili painting ’ s provenance with breuil ’ s ideas and revised the title to the ‘ white lady ’ of aouanrhet : in other paintings found a few days later in the same massif we were able to discern , from some characteristic features , an indication of egyptian influence . some features are , no doubt , not very marked in our `` white lady , '' still , all the same , some details as the curve of the breasts , led us to think that the picture may have been executed at a time when egyptian traditions were beginning to be felt in the tassili . foreign influence ? time and scholarship would reveal that the assignment of egyptian influence on the running horned woman was erroneous , and lhote the victim of a hoax : french members of his team made `` copies '' of egyptionized figures , passing them off as faithful reproductions of authentic tassili rock wall paintings . these fakes were accepted by lhote ( if indeed he knew nothing of the forgeries ) , and falsely sustained his belief in the possibility of foreign influence on central saharan rock art . breuil ’ s theories were likewise discredited : the myth of the `` white lady '' was rejected by every archaeologist of repute , and his promotion of foreign influence viewed as racist . yet breuil and lhote were not alone in finding it hard to believe that ancient africans discovered how to make art on their own , or to have developed artistic sensibilities . until quite recently many europeans maintained that art `` spread '' or was `` taken '' into africa , and , aiming to prove this thesis , anointed many works with classical sounding names and sought out similarities with early rock art in europe . although such vestiges of colonial thinking are today facing a reckoning , cases such as the `` white lady '' ( both of namibia and of tassili ) remind us of the perils of imposing cultural values from the outside . chronology while we have yet to learn how , and in what places , the practice of rock art began , no firm evidence has been found to show that african rock art—some ten million images across the continent—was anything other than a spontaneous initiative by early africans . scholars have estimated the earliest art to date to 12,000 or more years ago , yet despite the use of both direct and indirect dating techniques very few firm dates exist ( “ direct dating ” uses measurable physical and chemical analysis , such as radiocarbon dating , while “ indirect dating ” primarily uses associations from the archaeological context ) . in the north , where rock art tends to be quite diverse , research has focused on providing detailed descriptions of the art and placing works in chronological sequence based on style and content . this ordering approach results in useful classification and dating systems , dividing the tassili paintings and engravings into periods of concurrent and overlapping traditions ( the running horned woman is estimated to date to approximately 6,000 to 4,000 b.c.e.—placing it within the `` round head period '' ) , but offers little in the way of interpretation of the painting itself . advancing an interpretation of the running horned woman who was the running horned woman ? was she indeed a goddess , and her rock shelter some sort of sanctuary ? what does the image mean ? and why did the artist make it ? for so long the search for meaning in rock art was considered impossible—only recently have scholars endeavored to move beyond the mere description of images and styles , and , using a variety of interdisciplinary methods , make serious attempts to interpret the rock art of the central sahara . lhote recounted that the running horned woman was found on an isolated rock whose base was hollowed out into a number of small shelters that could not have been used as dwellings . this remote location , coupled with an image of marked pictorial quality—depicting a female with two horns on her head , dots on her body probably representing scarification , and wearing such attributes of the dance as armlets and garters—suggested to him that the site , and the subject of the painting , fell outside of the everyday . more recent scholarship has supported lhote ’ s belief in the painting ’ s symbolic , rather than literal , representation . as jitka soukopova has noted , `` hunter- gatherers were unlikely to wear horns ( or other accessories on the head ) and to make paintings on their whole bodies in their ordinary life . `` [ 1 ] rather , this female horned figure , her body adorned and decorated , found in one of the highest massifs in the tassili—a region is believed to hold special status due to its elevation and unique topology—suggests ritual , rite , or ceremony . but there is further work to be done to advance an interpretation of the running horned woman . increasingly scholars have studied rock shelter sites as a whole , rather than isolating individual depictions , and the shelter ’ s location relative to the overall landscape and nearby water courses , in order to learn the significance of various `` rock cities '' in both image making and image viewing . archaeological data from decorated pottery , which is a dated artistic tradition , is key in suggesting that the concept of art existed in the central sahara at the time of tassili rock art production . comparative studies with other rock art complexes , specifically the search for similarities in fundamental concepts in african religious beliefs , might yield the most fruitful approaches to interpretation . in other words , just as southern african rock studies have benefitted from tracing the beliefs and practices of the san people , so too may a study of tuareg ethnography shed light on the ancient rock art sites of the tassili . afterword : the threatened rock art of the central sahara tassili ’ s rock walls were commonly sponged with water in order to enhance the reproduction of its images , either in trace , sketch , or photograph . this washing of the rock face has had a devastating effect on the art , upsetting the physical , chemical , and biological balance of the images and their rock supports . many of the region ’ s subsequent visitors—tourists , collectors , photographers , and the next generation of researchers—all captivated by lhote ’ s `` discovery '' —have continued the practice of moistening the paintings in order to reveal them . today scholars report paintings that are severely faded while some have simply disappeared . in addition , others have suffered from irreversible damage caused by outright vandalism : art looted or stolen as souvenirs . in order to protect this valuable center of african rock art heritage , tassili n ’ ajjer was declared a national park in 1972 . it was classified as a world heritage site by unesco in 1982 and a biosphere reserve in 1986 . notes : [ 1 ] jitka soukopova , “ the earliest rock paintings of the central sahara : approaching interpretation , ” time and mind : the journal of archaeology , consciousness and culture 4 , no . 2 ( 2011 ) , p. 199 . essay by nathalie hager additional resources : african rock art : tassilinajjer – heilbrunn timeline of art history tassili n ’ ajjer – african world heritage sites tassili n ’ ajjer , unesco tara – trust for african rock art david coulson and alec campbell , african rock art : paintings and engravings on stone ( new york : harry n. abrams , inc. , 2001 ) . david coulson and alec campbell , “ rock art of the tassili n ajjer , algeria. ” adoranten , no . 1 ( 2010 ) , p. 115 . jeremy h. keenan , “ the lesser gods of the sahara. ” public archaeology 2 , no . 3 ( 2002 ) , pp . 131-50 . jean dominque lajoux , the rock paintings of tassili , translated by g. d. liversage ( london : thames and hudson , 1963 ) . henri lhote , the search for the tassili frescoes , translated by alan h. brodrick ( london : hutchinson & amp ; co. , 1959 ) . jitka soukopova , “ the earliest rock paintings of the central sahara : approaching interpretation , ” time and mind : the journal of archaeology , consciousness and culture 4 , no . 2 ( 2011 ) , pp . 193-216 .
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this ordering approach results in useful classification and dating systems , dividing the tassili paintings and engravings into periods of concurrent and overlapping traditions ( the running horned woman is estimated to date to approximately 6,000 to 4,000 b.c.e.—placing it within the `` round head period '' ) , but offers little in the way of interpretation of the painting itself . advancing an interpretation of the running horned woman who was the running horned woman ? was she indeed a goddess , and her rock shelter some sort of sanctuary ?
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were there any horned animals in that area that might have influenced the idea of horns on a person ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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} { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side .
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can these equations be as complex as 20 steps ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation .
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if addition and subtraction are of equal level in the order of operations , could n't another valid answer for 5 - 2 + 2 be 1 ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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\end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation .
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after solving the equation p-18=3 why they have written k=21 is a solution instead of p=21 is a solution ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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} { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side .
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how will these equations become useful in people 's every day lives ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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\\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ?
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check after applying the correct operation to each side , what is ttt ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ?
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how do we use pi in these questions ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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why are equation c and d the same but on different ends of the equals sign ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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} { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side .
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how do we use pi in these equations ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ?
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how do you get and keep the concept of knowing when to add and subtract from both sides ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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why is equation c and d pretty much the same thing ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ?
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why are there so many letters in algebra ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ?
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who came up with rules of math ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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is one step equation and two step equation have the same method ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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} { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side .
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can these equations be as complex as , even though it is unreasonable , 20 terms ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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if the problem is t-13=35 , why do you leave 13 blank if you need to add 13 to each side ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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} { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side .
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how do we use pi in these equations ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems .
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why are one of the qualifications to add a number to each side of the other number ?
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based on our understanding of the balance beam model , we know that we always have to do the same thing to both sides of an equation to keep it true . but how do we know what to do to both sides of the equation ? addition and subtraction are inverse operations inverse operations are opposite operations that undo or counteract each other . here 's an example of how subtraction is the inverse operation of addition : if we start with seven , add three , then subtract three , we get back to seven : $ 7 + 3 - 3 = 7 $ here 's an example of how addition is the inverse operation of subtraction : if we start with five , subtract two , then add two , we get back to five : $ 5 - 2 + 2 = 5 $ solving an addition equation using inverse operations let 's think about how we can solve for $ k $ in the following equation : $ \qquad k + 22 = 29 $ we want to get $ k $ by itself on the left hand side of the equation . so , what can we do to undo adding 22 ? we can subtract 22 because the inverse operation of addition is subtraction ! here 's how subtracting 22 from each side looks : $ \begin { align } k + 22 & amp ; = 29 \\ k + 22 \blued { - 22 } & amp ; = 29 \blued { - 22 } ~~~~~~~~~~\small\gray { \text { subtract 22 from each side . } } \\ k & amp ; = \greend { 7 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . it 's always a good idea to check our solution in the original equation to make sure we did n't make any mistakes : $ \qquad $ $ \begin { align } k +22 & amp ; = 29 \ \greend { 7 } +22 & amp ; \stackrel { \large ? } { = } 29\ 29 & amp ; = 29 \end { align } $ yes , $ k = \greend { 7 } $ is a solution ! solving a subtraction equation using inverse operations now let 's try to solve a slightly different type of equation : $ \qquad p - 18 = 3 $ we want to get $ p $ by itself on the left hand side of the equation . so , what can we do to cancel out subtracting 18 ? we can add 18 because the inverse operation of subtraction is addition ! here 's how adding 18 to each side looks : $ \begin { align } p - 18 & amp ; = 3 \\ p - 18 \blued { + 18 } & amp ; = 3 \blued { + 18 } ~~~~~~~~~~\small\gray { \text { add 18 to each side . } } \\ p & amp ; = \greend { 21 } ~~~~~~~~~~\small\gray { \text { simplify . } } \end { align } $ let 's check our work . $ \qquad $ $ \begin { align } p - 18 & amp ; = 3 \ \greend { 21 } - 18 & amp ; \stackrel { \large ? } { = } 3\ 3 & amp ; = 3 \end { align } $ yes , $ p = \greend { 21 } $ is a solution ! summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems . equation a : $ y + 6 = 52 $ equation b : $ 3 + y = 27 $ equation c : $ t - 13 = 35 $ equation d : $ 35 = t - 13 $
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summary of how to solve addition and subtraction equations cool , so we just solved an addition equation and a subtraction equation . let 's summarize what we did : type of equation | example | first step : - | : - : | : - : addition equation | $ k + 22 = 29 $ | subtract 22 from each side . subtraction equation | $ p - 18 = 3 $ | add 18 to each side . let 's try some problems .
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23+y=123 , 23 to the other side changing the sign ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent .
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who discovered / invented sohcahtoa ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent .
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so i know that arcsin ( sin ( x ) ) = x but ... what happens when you do arcsin ( x ) * sin ( x ) ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent .
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if the earth is really round , then why do commercial airliners going from south america to australia fly the long way over africa ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall !
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how to calculate the inverse function in a calculator ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent .
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sin ( 150 ) =0.5 and sin ( 30 ) =0.5 so for arcsin ( 0.5 ) which value should i take and why ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } }
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can you use pythagoras to find out the length of a missing side if you are given the other 2 lenghts ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below .
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what does the _m_ prefix mean before the angle ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations .
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what is the difference between trigonometric ratios and trigonometric functions ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall !
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is there any way to calculate arcsine or any inverse function in the iphone calculator ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below .
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what button do i use to find the approximate values of non-integer square roots ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations .
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do cosecant , secant , and cotangent also have inverse functions ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations .
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what are the uses of inverse trig functions ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent .
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why do we need to use sin ( -1 when we could just use sin ( ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent .
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according to the inverse function that has been said here , if i were to have a function , say , f ( x ) = 2x+3 , then f^-1 ( 2x+3 ) = x ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent .
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what does m refers to ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent .
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do the trees moving cause the wind to blow ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below .
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how do i find the trig ratio of a number without a calculator ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations .
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when and when not do we use the inverse of trig functions to calculate the answer , vice versa ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall !
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if it 's an inverse function , would n't tan be adjacent over opposite ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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\\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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how do you convert to degrees if your calculator measures in radians ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations .
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how do you solve an inverse problem in degrees ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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$ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator .
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how do you substitute the values ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ?
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how do you know when to use cosecant , secant , and cotangent instead of sine , cosine , and tangent ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ .
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in the triangle has opp of 35 and adj of 65 , so what is the measure of angle l ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ .
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i have question from in the previous lesson , called `` practice : solve for a side in right triangle '' from the triangle how do you know which one to divide or multiple ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent .
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is there any way to not use a calculator when finding out the sin , cos , tan , sin-1 , cos-1 , or tan-1 ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below .
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so im guessing for angle z you cant use 8.06/9 which would give you the arc sine any body know why ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent .
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how can you insert a square root sign ?
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let 's take a look at a new type of trigonometry problem . interestingly , these problems ca n't be solved with sine , cosine , or tangent . a problem : in the triangle below , what is the measure of angle $ l $ ? what we know : relative to $ \angle l $ , we know the lengths of the opposite and adjacent sides , so we can write : $ \tan ( l ) = \dfrac { \text { opposite } } { \text { adjacent } } = \dfrac { 35 } { 65 } $ but this does n't help us find the measure of $ \angle l $ . we 're stuck ! what we need : we need new mathematical tools to solve problems like these . our old friends sine , cosine , and tangent aren ’ t up to the task . they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations . each operation does the opposite of its inverse . the idea is the same in trigonometry . inverse trig functions do the opposite of the “ regular ” trig functions . for example : inverse sine $ ( \sin^ { -1 } ) $ does the opposite of the sine . inverse cosine $ ( \cos^ { -1 } ) $ does the opposite of the cosine . inverse tangent $ ( \tan^ { -1 } ) $ does the opposite of the tangent . in general , if you know the trig ratio but not the angle , you can use the corresponding inverse trig function to find the angle . this is expressed mathematically in the statements below . trigonometric functions input angles and output side ratios | |inverse trigonometric functions input side ratios and output angles : - : | : -| : - : $ \sin ( \theta ) =\dfrac { \text { opposite } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \sin^ { -1 } \left ( \dfrac { \text { opposite } } { \text { hypotenuse } } \right ) =\theta $ $ \cos ( \theta ) =\dfrac { \text { adjacent } } { \text { hypotenuse } } $ | $ \rightarrow $ | $ \cos^ { -1 } \left ( \dfrac { \text { adjacent } } { \text { hypotenuse } } \right ) =\theta $ $ \tan ( \theta ) =\dfrac { \text { opposite } } { \text { adjacent } } $ | $ \rightarrow $ | $ \tan^ { -1 } \left ( \dfrac { \text { opposite } } { \text { adjacent } } \right ) =\theta $ misconception alert ! the expression $ \sin^ { -1 } ( x ) $ is not the same as $ \dfrac { 1 } { \sin ( x ) } $ . in other words , the $ \small { -1 } $ is not an exponent . instead , it simply means inverse function . however , there is an alternate notation that avoids this pitfall ! the inverse sine can also be expressed as $ \arcsin $ , the inverse cosine as $ \arccos $ , and the inverse tangent as $ \arctan $ . this notation is common in computer programming languages , but not in mathematics . solving the introductory problem in the introductory problem , we were given the opposite and adjacent side lengths , so we can use inverse tangent to find the angle . $ \begin { align } { m\angle l } & amp ; =\tan^ { -1 } \left ( \dfrac { \text { } \blued { \text { opposite } } } { \text { } \maroonc { \text { adjacent } } \text { } } \right ) \quad\small { \gray { \text { define . } } } \\ m\angle l & amp ; =\tan^ { -1 } \left ( \dfrac { \blued { 35 } } { \maroonc { 65 } } \right ) \quad\small { \gray { \text { substitute values . } } } \\ m\angle l & amp ; \approx 28.30^\circ \quad\small { \gray { \text { evaluate with a calculator . } } } \end { align } $ now let 's try some practice problems .
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they take angles and give side ratios , but we need functions that take side ratios and give angles . we need inverse trig functions ! the inverse trigonometric functions we already know about inverse operations . for example , addition and subtraction are inverse operations , and multiplication and division are inverse operations .
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how to round for inverse sin ( 8/10 ) ?
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