Measuring Volume Packing cartons sit on a loading dock ready to be filled. (credit: “boxing day” by Erich Ferdinand/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Identify reasonable values for volume applications. Convert between like units of measures of volume. Convert between different unit values. Solve application problems involving volume. Volume is a measure of the space contained within or occupied by three-dimensional objects. It could be a box, a pool, a storage unit, or any other three-dimensional object with attributes that can be measured in the metric unit for distance–meters. For example, when purchasing an SUV, you may want to compare how many cubic units of cargo the SUV can hold. Cubic units indicate that three measures in the same units have been multiplied together. For example, to find the volume of a rectangular prism, you would multiply the length units by the width units and the height units to determine the volume in square units: 1  cm × 1  cm × 1  cm = 1  cm 3 Note that to accurately calculate volume, each of the measures being multiplied must be of the same units. For example, to find a volume in cubic centimeters, each of the measures must be in centimeters. The formula used to determine volume depends on the shape of the three-dimensional object. Here we will limit our discussions to the area to rectangular prisms like the one in Given this limitation, the basic formula for volume is: Volume = length ( l ) × width ( w ) × height ( h ) or V = l w h Rectangular Prism with Height ( h ) , Length ( l ) , and Width ( w ) Labeled. Reasonable Values for Volume Because volume is determined by multiplying three lengths, the magnitude of difference between different cubic units is exponential. In other words, while a meter is 100 times greater in length than a centimeter, a cubic meter, m 3 , is 100 × 100 × 100 ,  or  1,000,000 times greater in area than a cubic centimeter, cm 3 . This relationship between benchmark metric volume units is shown in the following table. Units Relationship Conversion Rate km 3 to m 3 km × km × km = km 3 1 km = 1,000 m 1,000  m × 1,000 m × 1,000 m = 1,000,000,000 m 3 1 km 3 = 1,000,000,000 m 3 m 3 to dm 3 m × m × m = m 3 1  m = 10  dm 10  dm × 10  dm × 10  dm = 1,000  dm 3 1 m 3 = 1,000 dm 3 dm 3 to cm 3 dm × dm × dm = dm 3 1  dm = 10  cm 10  cm × 10  cm × 10  cm = 1,000,000  cm 3 1 cm 3 = 1,000 dm 3 cm 3 to mm 3 cm × cm × cm = cm 3 1  cm = 10  mm 10  mm × 10  mm × 10  mm = 1,000  mm 3 1 cm 3 = 1,000 mm 3 To have an essential understanding of metric volume, you must be able to identify reasonable values for volume. When testing for reasonableness you should assess both the unit and the unit value. Only by examining both can you determine whether the given volume is reasonable for the situation. Determining Reasonable Values for Volume A grandparent wants to send cookies to their grandchild away at college. Which represents a reasonable value for the volume of a box to ship the cookies: 3,375 km 3 , 3,375 m 3 , or 3,375 cm 3 ? A volume of 3,375 km 3 is equivalent to a rectangular prism with dimensions of 15  km  × 15  km × 15  km , which is far too large for a shipping box. An area of 3,375 m 3 is equivalent to a surface of 15  m  × 15  m × 15  m , which is also too large. A reasonable value for the volume of the box is 3,375 cm 3 . Identifying Reasonable Values for Volume A food manufacturer is prototyping new packaging for one of its most popular products. Which represents a reasonable value for the volume of the box: 2 dm 3 , 2 cm 3 , or 2 mm 3 ? A decimeter is equal to 10 centimeters. A box with a volume of 2 dm 3 might have the dimensions 1  dm × 1  dm ×  2  dm , or 10  cm × 10  cm × 20  cm , which is reasonable. A box with a volume of 2 cm 3 or 2 mm 3 would be too small. Explaining Reasonable Values for Volume A farmer has a hay loft. They calculate the volume of the hayloft as 64 cm 3 . Does the calculation make sense? Explain your answer. No. Centimeters are used to determine smaller distances, such as the length of a pencil. A hayloft is more than 64 centimeters long, so a more reasonable unit of value would be m 2 . A volume of 64 m 3 can be calculated using the dimensions 4 meters by 4 meters by 4 meters, which are reasonable dimensions for a hayloft. So, a more reasonable value for the volume of the hayloft is 64 m 3 . Converting Like Units of Measures for Volume Just like converting units of measure for distance, you can convert units of measure for volume. However, the conversion factor, the number used to multiply or divide to convert from one volume unit to another, is different from the conversion factor for metric distance units. Recall that the conversion factor for volume is exponentially relative to the conversion factor for distance. The most frequently used conversion factors are illustrated in . Common Conversion Factors for Metric Volume Units Converting Like Units of Measure for Volume Using Multiplication A pencil case has a volume of 1,700 cm 3 . What is the volume in cubic millimeters? Use multiplication to convert from a larger metric volume unit to a smaller metric volume unit. To convert from cm 3 to mm 3 , multiply the value of the volume by 1,000. 1,700 × 1,000 = 1,700,000 The pencil case has a volume of 1,700,000 mm 3 . How to Convert Cubic Centimeters to Cubic Meters Converting Like Units of Measure for Volume Using Multi-Step Multiplication A shipping container has a volume of 33.2 m 3 . What is the volume in cubic centimeters? Use multiplication to convert a larger metric volume unit to a smaller metric volume unit. To convert from m 3 to cm 3 , first multiply the value of the volume by 1,000 to convert from m 3 to dm 3 , and then multiply again by 1,000 to convert from dm 3 to cm 3 . 33.2 × 1,000 × 1,000 = 33,200,000 The shipping container has a volume of 32,200,000 cm 3 . Converting Like Units of Measure for Volume Using Multi-Step Division A holding tank at the local aquarium has a volume of 22,712,000,000 cm 3 . What is the volume in cubic meters? indicates that when converting from a smaller metric volume unit to a larger metric volume unit you divide using the given conversion factor. To convert from cm 3 to m 3 , divide the value of the volume by 1,000 to first convert from cm 3 to dm 3 , then divide again to convert from dm 3 to m 3 . cm 3  to dm 3 :  22,712,000,000 1,000 = 22,712,000 dm 3  to m 3 :  22,712,000 1,000 = 22,712 The holding tank has a volume of 22,712 m 3 . Understanding Other Metric Units of Volume When was the last time you purchased a bottle of soda? Was the volume of the bottle expressed in cubic centimeters or liters? The liter (L) is a metric unit of capacity often used to express the volume of liquids. A liter is equal in volume to 1 cubic decimeter. A milliliter is equal in volume to 1 cubic centimeter. So, when a doctor orders 10 cc (cubic centimeters) of saline to be administered to a patient, they are referring to 10 mL of saline. The most frequently used factors for converting from cubic meters to liters are listed in . Relationships Between Metric Volume and Metric Capacity Units m 3 to L m 3 to mL 1  dm 3 = 1  L 1  dm 3 = 1,000  mL 1,000  cm 3 = 1  L 1  cm 3 = 1  mL 1,000,000  mm 3 = 1  L 1  mm 3 = 0.001  mL Converting Metric Units of Volume Converting Different Units of Measure for Volume A holding tank at the local aquarium has a volume of 22,712,000,000 cm 3 ? What is the capacity of the holding tank in liters? Use division to convert from cubic centimeters to liters. To determine the equivalent volume in liters, convert from cm 3 to L by dividing the value of the volume in cm 3 by 1,000. 22,712,000,000  cm 3 1,000 = 22,712,000  L The holding tank holds 22,712,000 L of water. Converting Different Units of Measure for Volume Using Multiplication An airplane used 150 m 3 of fuel to fly from New York to Hawaii. How many liters of fuel did the airplane use? Because 1 liter is equivalent to 1 cubic decimeter, use multiplication to convert from m 3 to dm 3 . Multiply the value of the volume by 1,000 to convert from m 3 to dm 3 . Because 1  dm 3 = 1  L , the resulting value is equivalent to the number of liters used. 150 × 1,000 = 150,000  dm 3 = 150,000  L The airplane used 150,000 liters of fuel. Converting Different Units of Measure for Volume Using Multi-Step Division How many liters can a pitcher with a volume of 8,000,000 mm 3 hold? Use division to convert from a smaller metric volume unit to a larger metric volume unit. To convert from mm 3 to dm 3 , Step 1: Divide by 1,000 to convert from mm 3 to cm 3 . Step 2: Divide again by 1,000 to convert from cm 3 to dm 3 . Step 3: Use the unit value to express the volume in terms of liters. 8,000,000 mm 3 1,000 = 8,000 cm 3 8,000 cm 3 1,000 = 8 dm 3 = 8 L The pitcher can hold 8 liters of liquid. Solving Application Problems Involving Volume Knowing the volume of an object lets you know just how much that object can hold. When making a bowl of punch you might want to know the total amount of liquid a punch bowl can hold. Knowing how many liters of gasoline a car’s tank can hold helps determine how many miles a car can drive on a full tank. Regardless of the application, understanding volume is essential to many every day and professional tasks. Using Volume to Solve Problems A cubic shipping carton’s dimensions measure 2 m × 2 m × 2 m . A company wants to fill the carton with smaller cubic boxes that measure 10  cm × 10  cm × 10  cm . How many of the smaller boxes will fit in each shipping carton? Step 1: Determine the volume of the shipping carton. 2  m × 2  m × 2  m = 8  m 3 Step 2: Use the appropriate conversion factor to convert the volume of the shipping carton from m 3 to cm 3 . 8  m 3 × 1,000 = 8,000  dm 3 8,000  dm 3 × 1,000 = 8,000,000  cm 3 Step 3: Determine the volume of the smaller boxes. 10  cm × 10  cm × 10  cm = 1,000  cm 3 Step 4: Divide the volume of the shipping carton, in cm 3 , by the volume of the smaller box, in cm 3 . 8,000,000  cm 3 1,000  cm 3 = 8,000 The shipping carton will hold 8,000 smaller boxes. Solving Volume Problems with Different Units A carton of juice measures 6 cm long, 6 cm wide and 20 cm tall. A factory produces 28,800 liters of orange juice each day. How many cartons of orange juice are produced each day? Step 1: Find the volume of the carton in cubic centimeters. 6  cm × 6  cm × 20  cm = 720  cm 3 Step 2: Convert the volume in cm 3 to liters. 1 cm 3 = 1 mL 720 cm 3 = 720 mL 720 mL 1,000 = 0.72 L Step 3: Divide the number of liters of orange juice produced each day by the volume of each carton. 28,800  L 0.72  L = 40,000 The factory produces 40,000 cartons of orange juice each day. Solving Complex Volume Problems A fish tank measures 60 cm long, 15 cm wide and 34 cm tall ( ). The tank is 25 percent full. How many liters of water are needed to completely fill the tank? Step 1: Determine the volume of the fish tank in cubic centimeters. 60  cm × 15  cm × 34  cm = 30,600  cm 3 Step 2: Convert the volume in cm 3 to volume in liters. 1 cm 3 = 1 mL 30,600 cm 3 = 30,600 mL 30,600 mL 1,000 = 30.6 L Step 3: Since the tank is 25 percent full, the tank is 75 percent empty. Convert 75 percent to its decimal equivalent. Multiply the total volume by 75 percent expressed in decimal form to determine how many liters of water are required to fill the tank. 75 % = 0.75 30.6 × 0.75 = 22.95 So, 22.95 liters of water are needed to fill the tank. How Does Shape Affect Volume? Take two large sheets of card stock. Roll one piece to tape the longer edges together to make a cylinder. Tape the cylinder to the other piece of card stock which serves as the base of the cylinder. Fill the cylinder to the top with cereal. Pour the cereal from the cylinder into a plastic storage or shopping bag. Remove the cylinder from the base and the tape from the cylinder. Re-roll the cylinder along the shorter edges a tape together. Attach the new cylinder to the base. Pour the cereal from the plastic bag into the cylinder. What do you observe? How does the shape of a container affect its volume? Check Your Understanding Key Terms volume cubic units Key Concepts Volume is a measure of the space contained within or occupied by three-dimensional objects. Volume is measured in cubic units; the base unit for volume in the metric system is cubic meters (m 3 ) The liter (L) is a metric unit of capacity but is often used to express the volume of liquids. One liter is equivalent to one cubic decimeter, which is the volume of a cube measuring 10 cm × 10 cm × 10 cm . Formulas To determine the volume of a rectangular prism: Volume = length ( l ) × width ( w ) × height ( h ) or V = l w h Rectangular Prism with Height ( h ) , Length ( l ) , and Width ( w ) Labeled You can convert between metric volume units and metric capacity units using the relationships shown in . Videos How to Convert Cubic Centimeters to Cubic Meters Converting Metric Units of Volume

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Measuring Weight Weight scale at the local Antigua market (credit: “Weight scale at the local Antigua market” by Lucía García González/Flickr, CC0 1.0 Public Domain Dedication) Learning Objectives After completing this section, you should be able to: Identify reasonable values for weight applications. Convert units of measures of weight. Solve application problems involving weight. In the metric system, weight is expressed in terms of grams or kilograms, with a kilogram being equal to 1,000 grams. A paper clip weighs about 1 gram. A liter of water weighs about 1 kilogram. In fact, in the same way that 1 liter is equal in volume to 1 cubic decimeter, the kilogram was originally defined as the mass of 1 liter of water. In some cases, particularly in scientific or medical settings where small amounts of materials are used, the milligram is used to express weight. At the other end of the scale is the metric ton (mt), which is equivalent to 1,000 kilograms. The average car weighs about 2 metric tons. Any discussion about metric weight must also include a conversation about mass . Scientifically, mass is the amount of matter in an object whereas weight is the force exerted on an object by gravity. The amount of mass of an object remains constant no matter where the object is. Identical objects located on Earth and on the moon will have the same mass, but the weight of the objects will differ because the moon has a weaker gravitational force than Earth. So, objects with the same mass will weigh less on the moon than on Earth. Since there is no easy way to measure mass, and since gravity is just about the same no matter where on Earth you go, people in countries that use the metric system often use the words mass and weight interchangeably. While scientifically the kilogram is only a unit of mass, in everyday life it is often used as a unit of weight as well. Reasonable Values for Weight To have an essential understanding of metric weight, you must be able to identify reasonable values for weight. When testing for reasonableness, you should assess both the unit and the unit value. Only by examining both can you determine whether the given weight is reasonable for the situation. Metric System: Units of Weight Identifying Reasonable Units for Weight Which is the more reasonable value for the weight of a newborn baby: 3.5 kg or 3.5 g? Using our reference weights, a baby weighs more than 3.5 paperclips, so 3.5 kilograms is a more reasonable value for the weight of a newborn baby. Determining Reasonable Values for Weight Which of the following represents a reasonable value for the weight of three lemons? 250 g, 2,500 g, or 250 kg? Because a kilogram is about 2.2 pounds, we can eliminate 250 kg as it is way too heavy. 2,500 grams is equivalent to 2.5 kilograms, or about five pounds, which is again, too heavy. So, a reasonable value for the weight of three lemons would be 250 grams. How Do You Measure the Weight of a Whale? It is impossible to weigh a living whale. Fredrik Christiansen from the Aarhus Institute of Advanced Studies in Denmark developed an innovative way to measure the weight of whales. Using images taken from a drone and computer modeling, the weight of a whale can be estimated with great accuracy. Using Drones to Weigh Whales? Explaining Reasonable Values for Weight The blue whale is the largest living mammal on Earth. Which of the following is a reasonable value for the weight of a blue whale: 149 g, 149 kg, or 149 mt? Explain your answer. A reasonable value for the weight of a blue whale is 149 metric tons. Both 149 g and 149 kg are much too small a value for the largest living mammal on Earth. Converting Like Units of Measures for Weight Just like converting units of measure for distance, you can convert units of measure for weight. The most frequently used conversion factors for metric weight are illustrated in . Common Conversion Factors for Metric Weight Units Converting Metric Units of Weight Using Multistep Division How many kilograms are in 24,300,000 milligrams? Use division to convert from a smaller metric weight unit to a larger metric weight unit. To convert from milligrams to kilograms, Step 1: Divide the value of the weight in milligrams by 1,000 to first convert from milligrams to grams. Step 2: Divide by 1,000 again to convert from grams to kilograms. 24,300,000  mg 1,000 = 24,300  g 24,300  g 1,000 = 24.3  kg So, 24,300,000 milligrams are equivalent to 24.3 kilograms. Converting Metric Units of Weight Using Multiplication The average ostrich weighs approximately 127 kilograms. How many grams does an ostrich weigh? Use multiplication to convert from a larger metric weight unit to a smaller metric weight unit. To convert from kilograms to grams, multiply the value of the weight by 1,000. 127  kg × 1,000 = 127,000  g The average ostrich weighs 127,000 grams. Converting Metric Units of Weight Using Multistep Multiplication How many milligrams are there in 0.025 kilograms? Use multiplication to convert from a larger metric weight unit to a smaller metric weight unit. To convert from kilograms to grams, Step 1: Multiply the value of the weight by 1,000. Step 2: Multiply the result by 1,000 to convert from grams to milligrams. 0.025  kg × 1,000 = 25  g 25 g × 1,000 = 25,000 mg So, 0.025 kilograms is equivalent to 25,000 milligrams. Metric Units of Mass: Convert mg, g, and kg Solving Application Problems Involving Weight From children’s safety to properly cooking a pie, knowing how to solve problems involving weight is vital to everyday life. Let’s review some ways that knowing how to work with metric weight can facilitate important decisions and delicious eating. Comparing Weights to Solve Problems The maximum weight for a child to safely use a car seat is 29 kilograms. If a child weighs 23,700 grams, can the child safely use the car seat? Step 1: Convert the child’s weight in grams to kilograms. 23,700  g ÷ 1,000 = 23.7  kg Step 2: Compare the two weights. 23.7  kg < 29  kg Yes, the child can safely use the car seat. Solving Multistep Weight Problems A recipe for scones calls for 350 grams of flour. How many kilograms of flour are required to make 4 batches of scones? Step 1: Multiply the grams of flour need by 4 to determine the total amount of flour needed. 350  g × 4 = 1,400  g Step 2: Convert from grams to kilograms. 1,400  g 1,000 = 1.4  kg So, 1.4 kilograms of flour are needed to make four batches of scones. Solving Complex Weight Problems The average tomato weighs 140 grams. A farmer needs to order boxes to pack and ship their tomatoes to local grocery stores. They estimate that this year’s harvest will yield 125,000 tomatoes. A box can hold 12 kilograms of tomatoes. How many boxes does the farmer need? Step 1: Determine the total estimated weight of the harvested tomatoes. 140  g × 125,000 = 17,500,000 g Step 2: Convert the total weight from grams to kilograms. 17,500,000  g ÷ 1,000 = 17,500  kg Step 3: Divide the weight of the tomatoes by the weight each box can hold. 17,500  kg ÷ 12  kg ≈ 1,458 So, the farmer will need to order 1,458 boxes. Check Your Understanding Key Terms mass Key Concepts Mass is the amount of matter in an object whereas weight is the force exerted on an object by gravity. The mass of an object never changes; the weight of an object changes depending on the force of gravity. An object with the same mass would weigh less on the moon than on Earth because the moon’s gravity is less than that of Earth. In the metric system, weight and mass are often used interchangeably and are expressed in terms in grams or kilograms. Videos Metric System: Units of Weight Using Drones to Weigh Whales? Metric Units of Mass: Convert mg, g, and kg Formulas You can convert between metric weight units using the conversion factors shown in .

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Measuring Temperature A thermometer that measures temperature in both customary and metric units. (credit: “Thermometer” by Jeff Djevdet/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Convert between Fahrenheit and Celsius. Identify reasonable values for temperature applications. Solve application problems involving temperature. When you touch something and it feels warm or cold, what is that really telling you about that substance? Temperature is a measure of how fast atoms and molecules are moving in a substance, whether that be the air, a stove top, or an ice cube. The faster those atoms and molecules move, the higher the temperature. In the metric system, temperature is measured using the Celsius (°C) scale. Because temperature is a condition of the physical properties of a substance, the Celsius scale was created with 100 degrees separating the point at which water freezes, 0 °C, and the point at which water boils, 100 °C. Scientifically, these are the points at which water molecules change from one state of matter to another—from solid (ice) to liquid (water) to gas (water vapor). When reading temperatures, it’s important to look beyond the degree symbol to determine which temperature scale the units express. For example, 13 °C reads “13 degrees Celsius,” indicating that the temperature is expressed using the Celsius scale, while 13 °F reads “13 degrees Fahrenheit,” indicating that the temperature is expressed using the Fahrenheit scale. Misconceptions About Temperature How Many Temperature Scales Are There? Did you know that in addition to Fahrenheit and Celsius, there is a third temperature scale widely used throughout the world? The Kelvin scale starts at absolute zero, the lowest possible temperature at which there is no heat energy present at all. It is primarily used by scientists to measure very high or very low temperatures when water is not involved. Converting Between Fahrenheit and Celsius Temperatures Understanding how to convert between Fahrenheit and Celsius temperatures is an essential skill in understanding metric temperatures. You likely know that below 32 °F means freezing temperatures and perhaps that the same holds true for 0 °C. While it may be difficult to recall that water boils at 212 °F, knowing that it boils at 100 °C is a fairly easy thing to remember. But what about all the temperatures in between? What is the temperature in degrees Celsisus on a scorching summer day? What about a cool autumn afternoon? If a recipe instructs you to preheat the oven to 350 °F, what Celsius temperature do you set the oven at? lists common temperatures on both scales, because we don’t use Celsius temperatures daily it’s difficult to remember them. Fortunately, we don’t have to. Instead, we can convert temperatures from Fahrenheit to Celsius and from Celsius to Fahrenheit using a simple algebraic expression. Common Temperatures The formulas used to convert temperatures from Fahrenheit to Celsius or from Celsius to Fahrenheit are outlined in . Fahrenheit to Celsius Celsius to Fahrenheit C = 5 9 ( F − 32 ) F = 9 5 C + 32 Temperature Conversion Formulas Converting Temperatures from Fahrenheit to Celsius A recipe calls for the oven to be set to 392 °F. What is the temperature in Celsius? Use the formula in to convert from Fahrenheit to Celsius. C = 5 9 ( F − 32 ) C = 5 9 ( 392 − 32 ) C = 5 9 ( 360 ) C = 200 So, 392 °F is equivalent to 200 °C. Converting Temperatures from Celsius to Fahrenheit On a sunny afternoon in May, the temperature in London was 20 °C. What was the temperature in degrees Fahrenheit? Use the formula in to convert from Celsius to Fahrenheit. F = 9 5 C + 32 F = 9 5 ( 20 ) + 32 F = 36 + 32 F = 68 The temperature was 68 °F. Comparing Temperatures in Celsius and Fahrenheit A manufacturer requires a vaccine to be stored in a refrigerator at temperatures between 36 °F and 46 °F. The refrigerator in the local pharmacy cools to 3 °C. Can the vaccine be stored safely in the pharmacy’s refrigerator? Use the formula in to convert from Celsius to Fahrenheit. F = 9 5 C + 32 F = 9 5 ( 3 ) + 32 F = 5.4 + 32 F = 37.4 Then, compare the temperatures. 36 F ° < 37.4 F ° < 46 F ° Yes. 37.4 °F falls within the acceptable range to store the vaccine, so it can be stored safely in the pharmacy’s refrigerator. Reasonable Values for Temperature While knowing the exact temperature is important in most cases, sometimes an approximation will do. When trying to assess the reasonableness of values for temperature, there is a quicker way to convert temperatures for an approximation using mental math. These simpler formulas are listed in . The formulas used to estimate temperatures from Fahrenheit to Celsius or from Celsius to Fahrenheit are outlined in . Fahrenheit to Celsius Celsius to Fahrenheit C = F − 30 2 F = 2 C + 30 Estimate Temperature Conversion Temperature Conversion Trick Using Benchmark Temperatures to Determine Reasonable Values for Temperatures Which is the more reasonable value for the temperature of a freezer? 5 °C or –5 °C? We know that water freezes at 0 °C. So, the more reasonable value for the temperature of a freezer is −5 °C, which is below 0 °C. At temperature of 5 °C is above freezing. Using Estimation to Determine Reasonable Values for Temperatures The average body temperature is generally accepted as 98.6 °F. What is a reasonable value for the average body temperature in degrees Celsius: 98.6 °C, 64.3 °C, or 34.3 °C? To estimate the average body temperature in degrees Celsius, subtract 30 from the temperature in degrees Fahrenheit, and divide the result by 2. ( 98.6 − 30 ) 2 = 68.6 2 = 34.3 A reasonable value for average body temperature is 34.3 °C. Using Conversion to Determine Reasonable Values for Temperatures Which is a reasonable temperature for storing chocolate: 28 °C, 18 °C, or 2 °C? Use the formula in to determine the temperature in degrees Fahrenheit. F = 9 5 C + 32 F = 9 5 ( 28 ) + 32 = 82.4 F = 9 5 ( 18 ) + 32 = 64.4 F = 9 5 ( 2 ) + 32 = 35.6 A temperature of 82.4 °F would be too hot, causing the chocolate to melt. A temperature of 35.6 °F is very close to freezing, which would affect the look and feel of the chocolate. So, a reasonable temperature for storing chocolate is 18 °C, or 64.4 °F. Solving Application Problems Involving Temperature Whether traveling abroad or working in a clinical laboratory, knowing how to solve problems involving temperature is an important skill to have. Many food labels express sizes in both ounces and grams. Most rulers and tape measures are two-sided with one side marked in inches and feet and the other in centimeters and meters. And while many thermometers have both Fahrenheit and Celsius scales, it really isn’t practical to pull out a thermometer when cooking a recipe that uses metric units. Let’s review at few instances where knowing how to fluently use the Celsius scale helps solve problems. Using Subtraction to Solve Temperature Problems The temperature in the refrigerator is 4 °C. The temperature in the freezer is 21 °C lower. What is the temperature in the freezer? Use subtraction to find the difference. 4 − 21 = − 17 So, the temperature in the freezer is −17 °C. Using Addition to Solve Temperature Problems A scientist was using a liquid that was 35 °C. They needed to heat the liquid to raise the temperature by 6 °C. What was the temperature after the scientist heated it? Use addition to find the new temperature. 35 + 6 = 41 The temperature of the liquid was 41 °C after the scientist heated it. Solving Complex Temperature Problems The optimum temperature for a chemical compound to develop its unique properties is 392 °F. When the heating process begins, the temperature of the compound is 20 °C. For safety purposes the compound can only be heated 9 °C every 15 minutes. How long until the compound reaches its optimum temperature? Step 1: Determine the optimum temperature in degrees Celsius using the formula in . C = 5 9 ( F − 32 ) C = 5 9 ( 392 − 32 ) C = 5 9 ( 360 ) C = 200 Step 2: Subtract the starting temperature. 200 C ° − 20 C ° = 180 C ° Step 3: Determine the number of 15-minute cycles needed to heat the compound to its optimum temperature. 180 ÷ 9 = 20 Step 4: Multiply the number of cycles needed by 15 minutes and convert the product to hours and minutes. 15  minutes × 9 = 135  minutes 135  minutes = 2  hours  15  minutes So, it will take 2 hours and 15 minutes for the compound to reach its optimum temperature. Learn the Metric System in 5 Minutes Check Your Understanding Key Terms temperature Key Concepts Temperature is a measure of how fast atoms and molecules are moving in a substance, whether that be the air, a stove top, or an ice cube. The faster those atoms and molecules move, the higher the temperature. In the metric system, temperature is measured using the Celsius (°C) scale. The Celsius scale was created with 100 degrees separating the point at which water freezes, 0 °C, and the point at which water boils, 100 °C. Scientifically, these are the points at which water molecules change from one state of matter to another—from solid (ice) to liquid (water) to gas (water vapor). Videos Misconceptions About Temperature Temperature Conversion Trick Learn the Metric System in 5 Minutes Formulas To convert temperature from Fahrenheit to Celsius: C = 5 9 ( F − 32 ) To convert temperature from Celsius to Fahrenheit: F = 9 5 C + 32 To estimate temperature from Fahrenheit to Celsius: C = F − 30 2 To estimate temperature from Celsius to Fahrenheit: F = 2 C + 30 Projects Cooking Take a favorite recipe that uses customary measures and convert the measures and cooking temperature to the metric system. Find a recipe that uses metric measures and convert the measures and cooking temperature to the U.S. Customary System of Measurement, using cups, tablespoons, or teaspoons as required. What did you observe? Was it easier to convert from one system to another? Which system allows for more precise measurements? What kitchen tools would you need in your kitchen if you used the metric system? Shopping Compare the average gas price in California to the average gas price in Puerto Rico. What conversions did you need to make to do the comparison? Do you think that the price of the gasoline is affected by the units in which it is sold? Sports What system of measurement is used for track and field events? Why do you think this system is used? What system of measurement is used for football? Why do you think this system is used? Research various sports records. Which units of measurement is used? What do you think influenced the unit of measure used? Chapter Review The Metric System Measuring Area Measuring Volume Measuring Weight Measuring Temperature Chapter Test

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Introduction The School of Athens by the Renaissance artist Raphael, painted between 1509 and 1511, depicts some of the greatest minds of ancient times. (credit: modification of work “School of Athens” by Raphael (1483–1520), Vatican Museums/Wikimedia, Public Domain) The painting The School of Athens presents great figures in history such as Plato, Aristotle, Socrates, Euclid, Archimedes, and Pythagoras. Other scientists are also represented in the painting. To the ancient Greeks, the study of mathematics meant the study of geometry above all other subjects. The Greeks looked for the beauty in geometry and did not allow their geometrical constructions to be “polluted” by the use of anything as practical as a ruler. They permitted the use of only two tools—a compass for drawing circles and arcs, and an unmarked straightedge to draw line segments. They would mark off units as needed. However, they never could be sure of what the units meant. For instance, how long is an inch? These mathematicians defined many concepts. The Greeks absorbed much from the Egyptians and the Babylonians (around 3000 BCE), including knowledge about congruence and similarity, area and volume, angles and triangles, and made it their task to introduce proofs for everything they learned. All of this historical wisdom culminated with Euclid in 300 BCE. Euclid (325–265 BC) is known as the father of geometry, and his most famous work is the 13-volume collection known as The Elements , which are said to be “the most studied books apart from the Bible.” Euclid brought together everything offered by the Babylonians, the Egyptians, and the more refined contributions by the Greeks, and set out, successfully, to organize and prove these concepts as he methodically developed formal theorems. This chapter begins with a discussion of the most basic geometric tools: the point, the line, and the plane. All other topics flow from there. Throughout the eight sections, we will talk about how to determine angle measurement and learn how to recognize properties of special angles, such as right angles and supplementary angles. We will look at the relationship of angles formed by a transversal, a line running through a set of parallel lines. We will explore the concepts of area and perimeter, surface area and volume, and transformational geometry as used in the patterns and rigid motions of tessellations. Finally, we will introduce right-angle trigonometry and explore the Pythagorean Theorem.

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Points, Lines, and Planes The lower right-hand corner of The School of Athens depicts a figure representing Euclid illustrating to students how to use a compass on a small chalkboard. (credit: modification of work “School of Athens” by Raphael (1483–1520), Vatican Museums/Wikimedia, Public Domain) Learning Objectives After completing this section, you should be able to: Identify and describe points, lines, and planes. Express points and lines using proper notation. Determine union and intersection of sets. In this section, we will begin our exploration of geometry by looking at the basic definitions as defined by Euclid. These definitions form the foundation of the geometric theories that are applied in everyday life. In The Elements , Euclid summarized the geometric principles discovered earlier and created an axiomatic system, a system composed of postulates. A postulate is another term for axiom, which is a statement that is accepted as truth without the need for proof or verification. There were no formal geometric definitions before Euclid, and when terms could not be defined, they could be described. In order to write his postulates, Euclid had to describe the terms he needed and he called the descriptions “definitions.” Ultimately, we will work with theorems, which are statements that have been proved and can be proved. Points and Lines The first definition Euclid wrote was that of a point. He defined a point as “that which has no part.” It was later expanded to “an indivisible location which has no width, length, or breadth.” Here are the first two of the five postulates, as they are applicable to this first topic: Postulate 1 : A straight line segment can be drawn joining any two points. Postulate 2 : Any straight line segment can be extended indefinitely in a straight line. Before we go further, we will define some of the symbols used in geometry in : Basic Geometric Symbols for Points and Lines From , we see the variations in lines, such as line segments, rays, or half-lines. What is consistent is that two collinear points (points that lie on the same line) are required to form a line. Notice that a line segment is defined by its two endpoints showing that there is a definite beginning and end to a line segment. A ray is defined by two points on the line; the first point is where the ray begins, and the second point gives the line direction. A half-line is defined by two points, one where the line starts and the other to give direction, but an open circle at the starting point indicates that the starting point is not part of the half-line. A regular line is defined by any two points on the line and extends infinitely in both directions. Regular lines are typically drawn with arrows on each end. Defining Lines For the following exercises, use this line ( ). Define D E ¯ . Define F . Define D F ↔ . Define E F ¯ . The symbol D E ¯ , two letters with a straight line above, refers to the line segment that starts at point D and ends at point E . The letter F alone refers to point F . The symbol D F ↔ , two letters with a line above containing arrows on both ends, refers to the line that extends infinitely in both directions and contains the points D and F . The symbol E F ¯ , two letters with a straight line above, refers to the line segment that starts at point E and ends at point F . There are numerous applications of line segments in daily life. For example, airlines working out routes between cities, where each city’s airport is a point, and the points are connected by line segments. Another example is a city map. Think about the intersection of roads, such that the center of each intersection is a point, and the points are connected by line segments representing the roads. See . Air Line Routes Determining the Best Route View the street map ( ) as a series of line segments from point to point. For example, we have vertical line segments A B ¯ , B C ¯ , and C D ¯ on the right. On the left side of the map, we have vertical line segments H I ¯ , F G . ¯ The horizontal line segments are H A ¯ , E I ¯ , I E ¯ , E B ¯ , F C ¯ , C F ¯ , and G B ¯ . There are two diagonal line segments, A E ¯ and E F ¯ . Assume that each location is on a corner and that you live next door to the library. Street Map Let’s say that you want to stop at the grocery store on your way home from school. Come up with three routes you might take to do your errand and then go home. In other words, name the three ways by the line segments in the order you would walk, and which way do you think would be the most efficient route? How about stopping at the library after school? Name four ways you might travel to the library and which way do you think is the most efficient? Suppose you need to go to the post office and the dry cleaners on your way home from school. Name three ways you might walk to do your errands and end up at home. Which way do you think is the most efficient way to walk, get your errands done, and go home? From school to the grocery store and home: first way A H ¯ , H I ¯ , I E ¯ , E F ¯ , F G ¯ ; second way A B ¯ , B E ¯ , E I ¯ , I E ¯ , E F ¯ , F G ¯ ; third way A E ¯ , E I ¯ , I E ¯ , E F ¯ , F G ¯ . It seems that the third way is the most efficient way. From school to the library: first way A E ¯ , E F ¯ , F G ¯ ; second way A B ¯ , B C ¯ , C D ¯ , D G ¯ ; third way A B ¯ , B E ¯ , E F ¯ , F G ¯ ; fourth way A B ¯ , B C ¯ , C F ¯ , F G ¯ . The first way should be the most efficient way. From school to the post office or dry cleaners to home: first way A E ¯ , E F ¯ , F C ¯ , C D ¯ , D G ¯ ; second way A B ¯ , B E ¯ , E F ¯ , F C ¯ , C D ¯ , D G ¯ ; third way A B ¯ , B C ¯ , C F ¯ , F G ¯ . The third way would be the most efficient way. Parallel Lines Parallel lines are lines that lie in the same plane and move in the same direction, but never intersect. To indicate that the line l 1 and the line l 2 are parallel we often use the symbol l 1 ∥ l 2 . The distance d between parallel lines remains constant as the lines extend infinitely in both directions. See . Parallel Lines Perpendicular Lines Two lines that intersect at a 90 ∘ angle are perpendicular lines and are symbolized by ⊥ . If l 1 and l 2 are perpendicular, we write l 1 ⊥ l 2 . When two lines form a right angle, a 90 ∘ angle, we symbolize it with a little square □ . See . Perpendicular Lines Identifying Parallel and Perpendicular Lines Identify the sets of parallel and perpendicular lines in . Drawing these lines on a grid is the best way to distinguish which pairs of lines are parallel and which are perpendicular. Because they are on a grid, we assume all lines are equally spaced across the grid horizontally and vertically. The grid also tells us that the vertical lines are parallel and the horizontal lines are parallel. Additionally, all intersections form a 90 ∘ angle. Therefore, we can safely say the following: A B ↔ ∥ C D ↔ , the line containing the points A and B is parallel to the line containing the points C and D . E F ↔ ∥ G H ↔ , the line containing the points E and F is parallel to the line containing the points G and H . A B ↔ ⊥ E F ↔ , the line containing the points A and B is perpendicular to the line containing the points E and F . We know this because both lines trace grid lines, and intersecting grid lines are perpendicular. We can also state that A B ↔ ⊥ G H ↔ ; the line containing the points A and B is perpendicular to the line containing the points G and H because both lines trace grid lines, which are perpendicular by definition. We also have C D ↔ ⊥ E F ↔ ; the line containing the points C and D is perpendicular to the line containing the points E and F because both lines trace grid lines, which are perpendicular by definition. Finally, we see that C D ↔ ⊥ G H ↔ ; the line containing the points C and D is perpendicular to the line containing the points G and H because both lines trace grid lines, which are perpendicular by definition. Defining Union and Intersection of Sets Union and intersection of sets is a topic from set theory that is often associated with points and lines. So, it seems appropriate to introduce a mini-version of set theory here. First, a set is a collection of objects joined by some common criteria. We usually name sets with capital letters. For example, the set of odd integers between 0 and 10 looks like this: A = { 1 , 3 , 5 , 7 , 9 } . When it involves sets of lines, line segments, or points, we are usually referring to the union or intersection of set. The union of two or more sets contains all the elements in either one of the sets or elements in all the sets referenced, and is written by placing this symbol ∪ in between each of the sets. For example, let set A = { 1 , 2 , 3 } , and let set B = { 4 , 5 , 6 } . Then, the union of sets A and B is A ∪ B = { 1 , 2 , 3 , 4 , 5 , 6 } . The intersection of two or more sets contains only the elements that are common to each set, and we place this symbol ∩ in between each of the sets referenced. For example, let’s say that set A = { 1 , 3 , 5 } , and let set B = { 5 , 7 , 9 } . Then, the intersection of sets A and B is A ∩ B = { 5 } . Defining Union and Intersection of Sets Use the line ( ) for the following exercises. Draw each answer over the main drawing. Find B D → ∩ C A ← . Find A B ¯ ∪ A D ¯ . Find A D ↔ ∪ B C ¯ . Find A D ↔ ∩ B C ¯ . Find B A ← ∩ C D ¯ . Find B D → ∩ C A ← . This is the intersection of the ray B D → and the ray C A ← . Intersection includes only the elements that are common to both lines. For this intersection, only the line segment B C ¯ is common to both rays. Thus, B D → ∩ C A ← = B C ¯ . Find A B ¯ ∪ A D ¯ . The problem is asking for the union of two line segments, A B ¯ and A D ¯ . Union includes all elements in the first line and all elements in the second line. Since A B ¯ is part of A D ¯ , A B ¯ ∪ A D ¯ = A D ¯ . Find A D ↔ ∪ B C ¯ . This is the union of the line A D ↔ with the line segment B C ¯ . As the line segment B C ¯ is included on the line A D ↔ , then the union of these two lines equals the line A D ↔ . Find A D ↔ ∩ B C ¯ . The intersection of the line A D ↔ with line segment B C ¯ is the set of elements common to both lines. In this case, the only element in common is the line segment B C ¯ . Find A B ← ∩ C D ¯ . This is the intersection of the ray A B ← and the line segment C D ¯ . Intersection includes elements common to both lines. There are no elements in common. The intersection yields the empty set, as shown in . Therefore, A B ← ∩ C D ¯ = ∅ . Planes A plane, as defined by Euclid, is a “surface which lies evenly with the straight lines on itself.” A plane is a two-dimensional surface with infinite length and width, and no thickness. We also identify a plane by three noncollinear points, or points that do not lie on the same line. Think of a piece of paper, but one that has infinite length, infinite width, and no thickness. However, not all planes must extend infinitely. Sometimes a plane has a limited area. We usually label planes with a single capital letter, such as Plane P , as shown in , or by all points that determine the edges of a plane. In the following figure, Plane P contains points A and B , which are on the same line, and point C , which is not on that line. By definition, P is a plane. We can move laterally in any direction on a plane. Plane P One way to think of a plane is the Cartesian coordinate system with the x -axis marked off in horizontal units, and y -axis marked off in vertical units. In the Cartesian plane, we can identify the different types of lines as they are positioned in the system, as well as their locations. See . Cartesian Coordinate Plane This plane contains points S , T , and R . Points T and R are colinear and form a line segment. Point S is not on that line segment. Therefore, this represents a plane. To give the location of a point on the Cartesian plane, remember that the first number in the ordered pair is the horizontal move and the second number is the vertical move. Point R is located at ( 4 , 2 ) ; point S is located at ( − 3 , 4 ) ; and point T is located at ( − 1 , − 1 ) . We can also identify the line segment as T R ¯ . Two other concepts to note: Parallel planes do not intersect and the intersection of two planes is a straight line. The equation of that line of intersection is left to a study of three-dimensional space. See . Parallel and Intersecting Planes To summarize, some of the properties of planes include: Three points including at least one noncollinear point determine a plane. A line and a point not on the line determine a plane. The intersection of two distinct planes is a straight line. Identifying a Plane For the following exercises, refer to Identify the location of points A , B , C , and D . Describe the line from point A to point D . Describe the line from point C containing point B . Does this figure represent a plane? Point A is located at ( 4 , 3 ) ; point B is located at ( − 4 , 1 ) ; point C is located at ( − 2 , − 3 ) ; point D is located at ( 1 , 1 ) . The line from point A to point D is a line segment A D . ¯ The line from point C containing point B is a ray C B → starting at point C in the direction of B . Yes, this figure represents a plane because it contains at least three points, points A and D form a line segment, and neither point B nor point C is on that line segment. Intersecting Planes Name two pairs of intersecting planes on the shower enclosure illustration ( ). The plane ABCD intersects plane CDEF , and plane CDEF intersects plane EFGH . Plato Part of a remarkable chain of Greek mathematicians, Plato (427–347 BC) is known as the teacher. He was responsible for shaping the development of Western thought perhaps more powerfully than anyone of his time. One of his greatest achievements was the founding of the Academy in Athens where he emphasized the study of geometry. Geometry was considered by the Greeks to be the “ultimate human endeavor.” Above the doorway to the Academy, an inscription read, “Let no one ignorant of geometry enter here.” The curriculum of the Academy was a 15-year program. The students studied the exact sciences for the first 10 years. Plato believed that this was the necessary foundation for preparing students’ minds to study relationships that require abstract thinking. The next 5 years were devoted to the study of the “dialectic.” The dialectic is the art of question and answer. In Plato’s view, this skill was critical to the investigation and demonstration of innate mathematical truths. By training young students how to prove propositions and test hypotheses, he created a culture in which the systematic process was guaranteed. The Academy was essentially the world’s first university and held the reputation as the ultimate center of learning for more than 900 years. Check Your Understanding Key Terms line segment plane union intersection parallel perpendicular Key Concepts Modern-day geometry began in approximately 300 BCE with Euclid’s Elements , where he defined the principles associated with the line, the point, and the plane. Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other. The union of two sets, A and B , contains all points that are in both sets and is symbolized as A ∪ B . The intersection of two sets A and B includes only the points common to both sets and is symbolized as A ∩ B .

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Angles This modern architectural design emphasizes sharp reflective angles as part of the aesthetic through the use of glass walls. (credit: “Société Générale @ La Défense @ Paris” by Images Guilhem Vellut/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Identify and express angles using proper notation. Classify angles by their measurement. Solve application problems involving angles. Compute angles formed by transversals to parallel lines. Solve application problems involving angles formed by parallel lines. Unusual perspectives on architecture can reveal some extremely creative images. For example, aerial views of cities reveal some exciting and unexpected angles. Add reflections on glass or steel, lighting, and impressive textures, and the structure is a work of art. Understanding angles is critical to many fields, including engineering, architecture, landscaping, space planning, and so on. This is the topic of this section. We begin our study of angles with a description of how angles are formed and how they are classified. An angle is the joining of two rays, which sweep out as the sides of the angle, with a common endpoint. The common endpoint is called the vertex . We will often need to refer to more than one vertex, so you will want to know the plural of vertex, which is vertices. In , let the ray A B → stay put. Rotate the second ray A C → in a counterclockwise direction to the size of the angle you want. The angle is formed by the amount of rotation of the second ray. When the ray A C → continues to rotate in a counterclockwise direction back to its original position coinciding with ray A B → , the ray will have swept out 360 ∘ . We call the rays the “sides” of the angle. Vertex and Sides of an Angle Classifying Angles Angles are measured in radians or degrees . For example, an angle that measures π radians, or 3.14159 radians, is equal to the angle measuring 180 ∘ . An angle measuring π 2 radians, or 1.570796 radians, measures 90 ∘ . To translate degrees to radians, we multiply the angle measure in degrees by π 180 . For example, to write 45 ∘ in radians, we have 45 ∘ ( π 180 ) = π 4 = 0.785398 radians. To translate radians to degrees, we multiply by 180 π . For example, to write 2 π radians in degrees, we have 2 π ( 180 π ) = 360 ∘ . Another example of translating radians to degrees and degrees to radians is 2 π 3 . To write in degrees, we have 2 π 3 ( 180 π ) = 120 ∘ . To write 30 ∘ in radians, we have 30 ∘ ( π 180 ) = π 6 . However, we will use degrees throughout this chapter. To translate an angle measured in degrees to radians, multiply by π 180 . To translate an angle measured in radians to degrees, multiply by 180 π . Several angles are referred to so often that they have been given special names. A straight angle measures 180 ∘ ; a right angle measures 90 ∘ ; an acute angle is any angle whose measure is less than 90 ∘ ; and an obtuse angle is any angle whose measure is between 90 ∘ and 180 ∘ . See . Classifying and Naming Angles An easy way to measure angles is with a protractor ( ). A protractor is a very handy little tool, usually made of transparent plastic, like the one shown here. Protractor (credit: modification of work “School drawing tools” by Marco Verch/Flickr, CC BY 2.0) With a protractor, you line up the straight bottom with the horizontal straight line of the angle. Be sure to have the center hole lined up with the vertex of the angle. Then, look for the mark on the protractor where the second ray lines up. As you can see from the image, the degrees are marked off. Where the second ray lines up is the measurement of the angle. Make sure you correctly match the center mark of the protractor with the vertex of the angle to be measured. Otherwise, you will not get the correct measurement. Also, keep the protractor in a vertical position. Notation Naming angles can be done in couple of ways. We can name the angle by three points, one point on each of the sides and the vertex point in the middle, or we can name it by the vertex point alone. Also, we can use the symbols ∠ or ∡ before the points. When we are referring to the measure of the angle, we use the symbol m ∡ . See . Naming an Angle We can name this angle ∡ B A C , or ∡ C A B , or ∡ A . Classifying Angles Determine which angles are acute, right, obtuse, or straight on the graph ( ). You may want to use a protractor for this one. Acute angles measure less than 90 ∘ . Obtuse angles measure between 90 ∘ and 180 ∘ . Right angles measure 90 ∘ . Straight angles measure 180 ∘ . ∠ E O F ∠ E O G ∠ F O G ∠ G O H ∠ F O H ∠ H O J ∠ H O K ∠ J O K ∠ K O L ∠ J O L ∠ E O J ∠ E O K ∠ F O J ∠ F O K ∠ F O L ∠ G O K ∠ G O L ∠ E O H ∠ H O L ∠ G O J ∠ E O L Most angles can be classified visually or by description. However, if you are unsure, use a protractor. Adjacent Angles Two angles with the same starting point or vertex and one common side are called adjacent angles. In , angle ∠ D B C is adjacent to ∠ C B A . Notice that the way we designate an angle is with a point on each of its two sides and the vertex in the middle. Adjacent Angles Supplementary Angles Two angles are supplementary if the sum of their measures equals 180 ∘ . In , we are given that m ∡ F B E = 35 ∘ , so what is m ∡ A B E ? These are supplementary angles. Therefore, because m ∡ A B F = 180 ∘ , and as 180 ∘ − 35 ∘ = 145 ∘ , we have m ∡ A B E = 145 ∘ . Supplementary Angles Solving for Angle Measurements and Supplementary Angles Solve for the angle measurements in . Step 1: These are supplementary angles. We can see this because the two angles are part of a horizontal line, and a horizontal line represents 180 ∘ . Therefore, the sum of the two angles equals 180 ∘ . Step 2: ( 32 x − 7 ) + ( 5 x + 2 ) = 180 37 x − 5 = 180 37 x = 185 x = 5 Step 3: Find the measure of each angle: 32 x − 7 = 32 ( 5 ) − 7 = 153 ∘ 5 x + 2 = 5 ( 5 ) + 2 = 27 ∘ Step 4: We check: 153 ∘ + 27 ∘ = 180 ∘ . Complementary Angles Two angles are complementary if the sum of their measures equals 90 ∘ . In , we have m ∡ A B C = 30 ∘ , and m ∡ A B D = 90 ∘ . What is the m ∡ C B D ? These are complementary angles. Therefore, because 90 ∘ − 30 ∘ = 60 ∘ , the ∡ C B D = 60 ° . Complementary Angles Solving for Angle Measurements and Complementary Angles Solve for the angle measurements in . We have that ( 9 x − 5 ) + 4 x + ( 7 x − 5 ) = 90 20 x = 100 x = 5 Then, m ∡ ( 9 x − 5 ) = 40 ∘ , m ∡ ( 4 x ) = 20 ∘ , and m ∡ ( 7 x − 5 ) = 30 ∘ . Vertical Angles When two lines intersect, the opposite angles are called vertical angles, and vertical angles have equal measure. For example, shows two straight lines intersecting each other. One set of opposite angles shows angle markers; those angles have the same measure. The other two opposite angles have the same measure as well. Vertical Angles Calculating Vertical Angles In , one angle measures 40 ∘ . Find the measures of the remaining angles. The 40-degree angle and ∠ 2 are vertical angles. Therefore, m ∡ 2 = 40 ∘ . Notice that ∡ 2 and ∡ 1 are supplementary angles, meaning that the sum of m ∡ 2 and m ∡ 1 equals 180 ∘ . Therefore, m ∡ 1 = 180 ∘ − 40 ∘ = 140 ∘ . Since ∡ 1 and ∡ 3 are vertical angles, then m ∡ 3 equals 140 ∘ . Transversals When two parallel lines are crossed by a straight line or transversal , eight angles are formed, including alternate interior angles, alternate exterior angles, corresponding angles, vertical angles, and supplementary angles. See . Angles 1, 2, 7, and 8 are called exterior angles, and angles 3, 4, 5, and 6 are called interior angles. Transversal Alternate Interior Angles Alternate interior angles are the interior angles on opposite sides of the transversal. These two angles have the same measure. For example, ∡ 3 and ∡ 6 are alternate interior angles and have equal measure; ∡ 4 and ∡ 5 are alternate interior angles and have equal measure as well. See . Alternate Interior Angles Alternate Exterior Angles Alternate exterior angles are exterior angles on opposite sides of the transversal and have the same measure. For example, in , ∡ 2 and ∡ 7 are alternate exterior angles and have equal measures; ∡ 1 and ∡ 8 are alternate exterior angles and have equal measures as well. Alternate Exterior Angles Corresponding Angles Corresponding angles refer to one exterior angle and one interior angle on the same side as the transversal, which have equal measures. In , ∡ 1 and ∡ 5 are corresponding angles and have equal measures; ∡ 3 and ∡ 7 are corresponding angles and have equal measures; ∡ 2 and ∡ 6 are corresponding angles and have equal measures; ∡ 4 and ∡ 8 are corresponding angles and have equal measures as well. Corresponding Angles Evaluating Space You live on the corner of First Avenue and Linton Street. You want to plant a garden in the far corner of your property ( ) and fence off the area. However, the corner of your property does not form the traditional right angle. You learned from the city that the streets cross at an angle equal to 150 ∘ . What is the measure of the angle that will border your garden? As the angle between Linton Street and First Avenue is 150 ∘ , the supplementary angle is 30 ∘ . Therefore, the garden will form a 30 ∘ angle at the corner of your property. Determining Angles Formed by a Transversal In given that angle 3 measures 40 ∘ , find the measures of the remaining angles and give a reason for your solution. m ∡ 2 = m ∡ 3 = 40 ∘ by vertical angles. ∡ 3 = m ∡ 7 by corresponding angles. m ∡ 7 = m ∡ 6 = 40 ∘ by vertical angles. m ∡ 1 = 180 − 40 = 140 ∘ by supplementary angles. m ∡ 4 = m ∡ 1 = 140 ∘ by vertical angles. m ∡ 8 = m ∡ 1 = 140 ∘ by alternate exterior angles. m ∡ 5 = m ∡ 8 = 140 ∘ by vertical angles. Measuring Angles Formed by a Transversal In given that angle 2 measures 23 ∘ , find the measure of the remaining angles and state the reason for your solution. m ∡ 2 = m ∡ 3 = 23 ∘ by vertical angles, because ∡ 2 and ∡ 3 are the opposite angles formed by two intersecting lines. m ∡ 1 = 157 ∘ by supplementary angles to m ∡ 2 or m ∡ 3. We see that ∡ 1 and ∡ 2 form a straight angle as does ∡ 1 and ∡ 3. A straight angle measures 180 ∘ , so 180 ∘ − 23 ∘ = 157 ∘ . m ∡ 4 = m ∡ 1 = 157 ∘ by vertical angles, because ∡ 4 and ∡ 1 are the two opposite angles formed by two intersecting lines. m ∡ 5 = m ∡ 1 = 157 ∘ by corresponding angles because they are the same angle formed by the transversal crossing two parallel lines, one exterior and one interior. m ∡ 8 = m ∡ 5 = 157 ∘ by vertical angles because ∡ 8 and ∡ 5 are the two opposite angles formed by two intersecting lines. m ∡ 7 = m ∡ 2 = 23 ∘ by alternate exterior angles because, like vertical angles, these angles are the opposite angles formed by the transversal intersecting two parallel lines. m ∡ 6 = m ∡ 7 = 23 ∘ by vertical angles because these are the opposite angles formed by two intersecting lines. Finding Missing Angles Find the measures of the angles 1, 2, 4, 11, 12, and 14 in and the reason for your answer given that l 1 and l 2 are parallel. m ∡ 12 = 118 ∘ , supplementary angles m ∡ 14 = 118 ∘ , vertical angles m ∡ 11 = 62 ∘ , vertical angles m ∡ 4 = 62 ∘ , corresponding angles m ∡ 1 = 62 ∘ , vertical angles m ∡ 2 = 56 ∘ , supplementary angles The Number 360 Did you ever wonder why there are 360 ∘ in a circle? Why not 100 ∘ or 500 ∘ ? The number 360 was chosen by Babylonian astronomers before the ancient Greeks as the number to represent how many degrees in one complete rotation around a circle. It is said that they chose 360 for a couple of reasons: It is close to the number of days in a year, and 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, … Check Your Understanding Key Terms vertex right angle acute angle obtuse angle straight angle complementary supplementary Key Concepts Angles are classified as acute if they measure less than 90 ∘ , obtuse if they measure greater than 90 ∘ and less than 180 ∘ , right if they measure exactly 90 ∘ , and straight if they measure exactly 180 ∘ . If the sum of angles equals 90 ∘ , they are complimentary angles. If the sum of angles equals 180 ∘ , they are supplementary. A transversal crossing two parallel lines form a series of equal angles: alternate interior angles, alternate exterior angles, vertical angles, and corresponding angles Formula To translate an angle measured in degrees to radians, multiply by π 180 . To translate an angle measured in radians to degrees, multiply by 180 π .

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Triangles The appearance of triangles in buildings is part of modern-day architectural design. (credit: "Inside Hallgrímskirkja church, Reykjavik, Iceland" by O Palsson/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Identify triangles by their sides. Identify triangles by their angles. Determine if triangles are congruent. Determine if triangles are similar. Find the missing side of similar triangles. How were the ancient Greeks able to calculate the radius of Earth? How did soldiers gauge their target? How was it possible centuries ago to estimate the height of a sail at sea? Triangles have always played a significant role in how we find heights of objects too high to measure or distances between objects too far away to calculate. In particular, the concept of similar triangles has countless applications in the real world, and we shall explore some of those applications in this section. Technology has given us instruments that allow us to find measurements of distant objects with little effort. However, it is all based on the properties of triangles discovered centuries ago. In this section, we will explore the various types of triangles and their special properties, as well as how to measure interior and exterior angles. We will also explore congruence theorems and similarity. Identifying Triangles Joining any three noncollinear points with line segments produces a triangle. For example, given points A , B , and C , connected by the line segments A B ¯ , B C , ¯ and A C ¯ , we have a triangle, as shown in . Triangle Triangles are classified by their angles and their sides. All angles in an acute triangle measure < 90 ∘ . One of the angles in a right triangle measures 90 ∘ , symbolized by □. One angle in an obtuse triangle measures between 90 ∘ and 180 ∘ . Sides that have equal length are indicated by the same hash marks. illustrates the shapes of the basic triangles, their names, and their properties. A few other facts to remember as we move forward: The points where the line segments meet are called the vertices (plural for vertex ). We often refer to sides of a triangle by the angle they are opposite. In other words, side a is opposite angle A , side b is opposite angle B , and side c is opposite angle C . Types of Triangles We want to add a special note about right triangles here, as they are referred to more than any other triangle. The side opposite the right angle is its longest side and is called the hypotenuse , and the sides adjacent to the right angle are called the legs. One of the most important properties of triangles is that the sum of the interior angles equals 180 ∘ . Euclid discovered and proved this property using parallel lines. The completed sketch is shown in . Sum of Interior Angles This is how the proof goes: Step 1: Start with a straight line A B ↔ and a point C not on the line. Step 2: Draw a line through point C parallel to the line A B ↔ . Step 3: Construct two transversals (a line crossing the parallel lines), one angled to the right and one angled to the left, to intersect the parallel lines. Step 4: Because of the property that alternate interior angles inside parallel lines are equal, we have that m ∡ 2 = m ∢ 1 and m ∡ 3 = m ∡ 4. Step 5: Notice that m ∡ 2 + m ∡ 5 + m ∡ 4 = 180 ∘ by the straight angle property. Step 6: Therefore, by substitution, m ∡ 1 = m ∡ 2 , and m ∡ 3 = m ∡ 4 , we have that m ∡ 1 + m ∡ 3 + m ∢ 5 = 180 ∘ . Therefore, the sum of the interior angles of a triangle = 180 ∘ . Finding Measures of Angles Inside a Triangle Find the measure of each angle in the triangle shown ( ). We know that the sum of the angles must equal 180 ∘ . Step 1: As the sum of the interior angles equals 180 ∘ , we can use algebra to find the measures: x + ( x + 17 ) + ( 3 x − 62 ) = 180 5 x − 45 = 180 5 x = 225 x = 225 5 = 45 Step 2: Now that we have the value of x , we can substitute 45 into the other two expressions to find the measure of those angles: ( x + 17 ) = 45 + 17 = 62 ∘ ( 3 x − 62 ) = 3 ( 45 ) − 62 = 135 − 62 = 73 ∘ Step 3: Then, m ∡ x = 45 ∘ , m ∡ ( x + 17 ) = 62 ∘ , and m ∡ ( 3 x − 62 ) = 73 ∘ . Finding Angle Measures Find the measure of angles numbered 1–5 in . The m ∡ 1 = 119 ∘ because it is supplementary with the unknown angle of the adjacent triangle. The unknown angle measures 61 ∘ . The m ∡ 2 = 61 ∘ because of vertical angles. The m ∡ 5 = 59 ∘ because the angle that is supplementary to the 134 ∘ measures 46 ∘ , and angle 5 is the unknown angle in that triangle. The m ∡ 4 = 59 ∘ by vertical angles. Finally, m ∡ 3 = 60 ∘ , as it is the third angle in the triangle with angles measuring 59 ∘ and 61 ∘ . Congruence If two triangles have equal angles and their sides lengths are equal, the triangles are congruent. In other words, if you can pick up one triangle and place it on top of the other triangle and they coincide, even if you have to rotate one, they are congruent. Determining If Triangles Are Congruent In , is the triangle ABC congruent to triangle DEF ? Triangle ABC is congruent to triangle DEF . Angles A and C are congruent to angles D and F , which implies that angle B is congruent to angle E . Side AB is congruent to side DE , and side CB is congruent to side FE , which implies that side AC is congruent to side DF . The Congruence Theorems The following theorems are tools you can use to prove that two triangles are congruent. We use the symbol ≅ to define congruence. For example, Δ A B C ≅ Δ D E F . Side-Side-Side (SSS). If three sides of one triangle are equal to the corresponding sides of the second triangle, then the triangles are congruent. See . Side-Side-Side (SSS) We have that D F ¯ ≅ R T ¯ , E F ¯ ≅ S T ¯ , and DE ¯ ≅ RS ¯ , then Δ D E F ≅ Δ R S T . Side-Angle-Side (SAS). If two sides of a triangle and the angle between them are equal to the corresponding two sides and included angle of the second triangle, then the triangles are congruent. See . We see that A B ¯ ≅ A ′ B ′ ¯ and B C ¯ ≅ B ′ C ′ ¯ , m ∡ B = m ∡ B ′ , then Δ A B C ≅ Δ A ′ B ′ C ′ . Side-Angle-Side (SAS) Angle-Side-Angle (ASA). If two angles and the side between them in one triangle are congruent to the two corresponding angles and the side between them in a second triangle, then the two triangles are congruent. See . Notice that m ∡ A ≅ m ∡ F , and m ∡ C ≅ m ∡ D , AC ¯ ≅ DF ¯ , then Δ ABC ≅ Δ DEF . Angle-Side-Angle (ASA) Angle-Angle-Side (AAS). If two angles and a nonincluded side of one triangle are congruent to two angles and the nonincluded corresponding side of a second triangle, then the triangles are congruent. See . We see that m ∡ X ≅ m X ′ , m ∡ Z ≅ m ∡ Z ′ , and X Y ¯ ≅ X ′ Y ′ ¯ , then Δ X Y Z ≅ Δ X ′ Y ′ Z ′ . Angle-Angle-Side (AAS) Identifying Congruence Theorems What congruence theorem is illustrated in ? AAS: Two angles and a non-included side in one triangle are congruent to the corresponding angles and side in the second triangle. Determining the Congruence Theorem What congruence theorem is illustrated in ? The SSS theorem. Similarity If two triangles have the same angle measurements and are the same shape but differ in size, the two triangles are similar. The lengths of the sides of one triangle will be proportional to the corresponding sides of the second triangle. Note that a single fraction a b is called a ratio, but two fractions equal to each other is called a proportion, such as a b = c d . This rule of similarity applies to all shapes as well as triangles. Another way to view similarity is by applying a scaling factor , which is the ratio of corresponding measurements between an object or representation of the object, to an image that produces the second, similar image. For example, why are the two images in are similar? These two images have the same proportions between elements. Therefore, they are similar. Similarity Determining If Triangles Are Similar Are the two triangles shown in similar? Step 1: We will look at the proportions within each triangle. In triangle α (alpha), the side opposite the 57 ∘ angle measures 7, and the side opposite the 33 ∘ angle measures 4. Then, the measures of the corresponding sides in triangle β (beta) measures 3.5 and 2, respectively. We have 4 7 = 0.5714 2 3.5 = 0.5714. This is the proportion 4 7 = 2 3.5 . The scaling factor is 0.5714. Step 2: Let’s try another correspondence. In triangle α , the hypotenuse measures 8.06 and the side opposite the 57 ∘ angle measures 7. In triangle β , the hypotenuse measure 4.03 and the side opposite the 57 ∘ angle measures 3.5. We have 7 8.06 = 0.8685 3.5 4.03 = 0.8685. Step 3: Now, let’s look at the proportions between triangle α and triangle β . The side measuring 2 in triangle β corresponds to the side measuring 4 in triangle α , the side measuring 3.5 in triangle β corresponds to the side measuring 7 in triangle α , and the hypotenuse in triangle β corresponds to the hypotenuse in triangle α . We have 2 4 = 0.5 3.5 7 = 0.5 4.03 8.06 = 0.5 Thus, the corresponding angles are equal and the proportions between each pair of corresponding sides equals 0.5. In other words, the scaling factor is 0.5. Therefore, the triangles are similar. Proving Similarity In , is triangle δ (delta) similar to triangle ε (epsilon)? Find the lengths of sides x and y as part of your answer. We can see that all three angles in triangle δ are equal to the corresponding angles in triangle ε . That is enough to determine similarity. However, we want to find the values of x and y to prove similarity. Step 1: We have to do is set up the proportions between the corresponding sides. We have the side that measures 2.375 in triangle δ corresponding to the side measuring 1.069 in triangle ε . We have the hypotenuse/side in triangle δ measuring 6 corresponding to the hypotenuse/side labeled y in triangle ε . And, finally, the side labeled x in triangle δ corresponds to the side measuring 2.475 in triangle ε . Each proportion should be equal. We start with the proportion of the shorter sides. Thus 1.069 2.375 = 0.45 Step 2: We solve for y using the first proportion. Set the two ratios equal to each other, cross-multiply, and solve for y . We have: 1.069 2.375 = y 6 ( 6 ) ( 1.069 ) = ( 2.375 ) ( y ) 6.414 = 2.375 y 6.414 2.375 = 2.7 = y So, y = 2.7 . Step 3: Checking that length in the proportion factor of 0.45, we have: y 6 = 2.7 6 = 0.45 Step 4: Solving for x , we will use the same proportion we used to solve for y . We have: 1.069 2.375 = 2.475 x 1.069 ( x ) = 2.475 ( 2.375 ) 1.069 ( x ) = 5.878 x = 5.878 1.069 = 5.5 2.475 x = 2.475 5.5 = 0.45 Step 5: We test the proportions. We have the following: 2.7 6 = 2.475 5.5 = 1.069 2.375 = 0.45 The proportions are all equal. Therefore, we have proven the property of similarity between triangle δ and triangle ε . Applying Similar Triangles A person who is 5 feet tall is standing 50 feet away from the base of a tree ( ). The tree casts a 57-foot shadow. The person casts a 7-foot shadow. What is the height of the tree? The bigger triangle includes a tree at side x and the smaller triangle includes the person at the side labeled 5 ft. These two triangles are similar because the smaller triangle fits inside the larger triangle at the smallest angle. It would fit inside the larger triangle at either of the other two angles as well. That all angles are equal is one of the criteria for similar triangles, so we can solve using proportions: 5 7 = x 57 5 ( 57 ) = 7 x 285 7 = x = 40.7 The tree is 40.7 feet tall. Finding Missing Lengths At a certain time of day, a radio tower casts a shadow 180 feet long ( ). At the same time, a 9-foot truck casts a shadow 15 feet long. What is the height of the tower? These are similar triangles and the problem can be solved by using proportions: x 180 = 9 15 9 ( 180 ) = 15 x 1620 = 15 x 108 = x The height of the tower is 108 ft. Thales of Miletus Thales of Miletus, sixth century BC, is considered one of the greatest mathematicians and philosophers of all time. Thales is credited with being the first to discover that the two angles at the base of an isosceles triangle are equal, and that the two angles formed by intersecting lines are equal—that is, vertical or opposite angles, are equal. Thales is also known for devising a method for measuring the height of the pyramids by similar right triangles. shows his method. He measured the length of the shadow cast by the pyramid at the precise time when his own shadow ended at the same place. Thales and Similarity He equated the vertical height of the pyramid with his own height; the horizontal distance from the pyramid to the tip of its shadow with the distance from himself and the tip of his own shadow; and finally, the length of the shadow cast off the top of the pyramid with length of his own shadow cast off the top of his head. Using proportions, as shown in , he essentially discovered the properties of similarity for right triangles. That is, △ A B C is similar to △ A ′ B ′ C ′ . Note that to be similar, all corresponding angles between the two triangles must be equal, and the proportions from one side to another side within each triangle, as well as the proportions of the corresponding sides between the two triangles must be equal. Thales is also credited with discovering a method of determining the distance of a ship from the shoreline. Here is how he did it, as illustrated in . Thales and Similar Triangles Thales walked along the shoreline pointing a stick at the ship until it formed a 90 ∘ angle to the shore. Then he walked along the shot and placed the stick in the ground at point C . He continued walking until he reached point D . Then, he turned and walked away from the shore at a 90 ∘ angle until the stick he placed in the ground at point C lined up with the ship, point E . This is how he created similar triangles and estimated the distance of the ship to the shore by using proportions. Check Your Understanding Key Terms acute obtuse isosceles equilateral hypotenuse congruence similarity scaling factor Key Concepts The sum of the interior angles of a triangle equals 180 ∘ . Two triangles are congruent when the corresponding angles have the same measure and the corresponding side lengths are equal. The congruence theorems include the following: SAS, two sides and the included angle of one triangle are congruent to the same in a second triangle; ASA, two angles and the included side of one triangle are congruent to the same in a second triangle; SSS, all three side lengths of one triangle are congruent to the same in a second triangle; AAS, two angles and the non-included side of one triangle are congruent to the same in a second triangle. Two shapes are similar when the proportions between corresponding angles, sides or features of two shapes are equal, regardless of size.

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Polygons, Perimeter, and Circumference Geometric patterns are often used in fabrics due to the interest the shapes create. (credit: "Triangles" by Brett Jordan/Flickr, CC BY 2.0) Learning Objectives After completing this section, you should be able to: Identify polygons by their sides. Identify polygons by their characteristics. Calculate the perimeter of a polygon. Calculate the sum of the measures of a polygon’s interior angles. Calculate the sum of the measures of a polygon’s exterior angles. Calculate the circumference of a circle. Solve application problems involving perimeter and circumference. In our homes, on the road, everywhere we go, polygonal shapes are so common that we cannot count the many uses. Traffic signs, furniture, lighting, clocks, books, computers, phones, and so on, the list is endless. Many applications of polygonal shapes are for practical use, because the shapes chosen are the best for the purpose. Modern geometric patterns in fabric design have become more popular with time, and they are used for the beauty they lend to the material, the window coverings, the dresses, or the upholstery. This art is not done for any practical reason, but only for the interest these shapes can create, for the pure aesthetics of design. When designing fabrics, one has to consider the perimeter of the shapes, the triangles, the hexagons, and all polygons used in the pattern, including the circumference of any circular shapes. Additionally, it is the relationship of one object to another and experimenting with different shapes, changing perimeters, or changing angle measurements that we find the best overall design for the intended use of the fabric. In this section, we will explore these properties of polygons, the perimeter, the calculation of interior and exterior angles of polygons, and the circumference of a circle. Identifying Polygons A polygon is a closed, two-dimensional shape classified by the number of straight-line sides. See for some examples. We show only up to eight-sided polygons, but there are many, many more. Types of Polygons If all the sides of a polygon have equal lengths and all the angles are equal, they are called regular polygons . However, any shape with sides that are line segments can classify as a polygon. For example, the first two shapes, shown in and , are both pentagons because they each have five sides and five vertices. The third shape is a hexagon because it has six sides and six vertices. We should note here that the hexagon in is a concave hexagon, as opposed to the first two shapes, which are convex pentagons. Technically, what makes a polygon concave is having an interior angle that measures greater than 180 ∘ . They are hollowed out, or cave in, so to speak. Convex refers to the opposite effect where the shape is rounded out or pushed out. Polygons While there are variations of all polygons, quadrilaterals contain an additional set of figures classified by angles and whether there are one or more pairs of parallel sides. See . Types of Quadrilaterals Identifying Polygons Identify each polygon. This shape has six sides. Therefore, it is a hexagon. This shape has four sides, so it is a quadrilateral. It has two pairs of parallel sides making it a parallelogram. This shape has eight sides making it an octagon. This is an equilateral triangle, as all three sides are equal. This is a rhombus; all four sides are equal. This is a regular octagon, eight sides of equal length and equal angles. Determining Multiple Polygons What polygons make up ? Shapes 1 and 5 are hexagons; shapes 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 15, and 16 are triangles; shapes 8 and 17 are parallelograms; and shape 11 is a trapezoid. Perimeter Perimeter refers to the outside measurements of some area or region given in linear units. For example, to find out how much fencing you would need to enclose your backyard, you will need the perimeter. The general definition of perimeter is the sum of the lengths of the sides of an enclosed region. For some geometric shapes, such as rectangles and circles, we have formulas. For other shapes, it is a matter of just adding up the side lengths. A rectangle is defined as part of the group known as quadrilaterals, or shapes with four sides. A rectangle has two sets of parallel sides with four angles. To find the perimeter of a rectangle, we use the following formula: The formula for the perimeter P of a rectangle is P = 2 L + 2 W , twice the length L plus twice the width W . For example, to find the length of a rectangle that has a perimeter of 24 inches and a width of 4 inches, we use the formula. Thus, 24 = 2 l + 2 ( 4 ) = 2 l + 8 24 − 8 = 2 l 16 = 2 l 8 = l The length is 8 units. The perimeter of a regular polygon with n sides is given as P = n ⋅ s . For example, the perimeter of an equilateral triangle, a triangle with three equal sides, and a side length of 7 cm is P = 3 ( 7 ) = 21 cm . Finding the Perimeter of a Pentagon Find the perimeter of a regular pentagon with a side length of 7 cm ( ). A regular pentagon has five equal sides. Therefore, the perimeter is equal to P = 5 ( 7 ) = 35 cm . Finding the Perimeter of an Octagon Find the perimeter of a regular octagon with a side length of 14 cm ( ). A regular octagon has eight sides of equal length. Therefore, the perimeter of a regular octagon with a side length of 14 cm is P = 8 ( 14 ) = 112 cm . Sum of Interior and Exterior Angles To find the sum of the measurements of interior angles of a regular polygon, we have the following formula. The sum of the interior angles of a polygon with n sides is given by S = ( n − 2 ) 180 ∘ . For example, if we want to find the sum of the interior angles in a parallelogram, we have S = ( 4 − 2 ) 180 ∘ = 2 ( 180 ) = 360 ∘ . Similarly, to find the sum of the interior angles inside a regular heptagon, we have S = ( 7 − 2 ) 180 ∘ = 5 ( 180 ) = 900 ∘ . To find the measure of each interior angle of a regular polygon with n sides, we have the following formula. The measure of each interior angle of a regular polygon with n sides is given by a = ( n − 2 ) 180 ∘ n . For example, find the measure of an interior angle of a regular heptagon, as shown in . We have a = ( 7 − 2 ) 180 ∘ 7 = 128.57 ∘ . Interior Angles Calculating the Sum of Interior Angles Find the measure of an interior angle in a regular octagon using the formula, and then find the sum of all the interior angles using the sum formula. An octagon has eight sides, so n = 8 . Step 1: Using the formula a = ( n − 2 ) 180 ∘ 8 : a = ( 8 − 2 ) 180 ∘ 8 = ( 6 ) 180 ∘ 8 = 135 ∘ . So, the measure of each interior angle in a regular octagon is 135 ∘ . Step 2: The sum of the angles inside an octagon, so using the formula: S = ( n − 2 ) 180 ∘ = ( 8 − 2 ) 180 ∘ = 6 ( 180 ) = 1,080 ∘ . Step 3: We can test this, as we already know the measure of each angle is 135 ∘ . Thus, 8 ( 135 ∘ ) = 1,080 ∘ . Calculating Interior Angles Use algebra to calculate the measure of each interior angle of the five-sided polygon ( ). Step 1: Let us find out what the total of the sum of the interior angles should be. Use the formula for the sum of the angles in a polygon with n sides: S = ( n − 2 ) 180 ∘ . So, S = ( 5 − 2 ) 180 ∘ = 540 ∘ . Step 2: We add up all the angles and solve for x : 5 ( x + 7 ) + 120 + ( 6 x + 25 ) + 5 ( 2 x + 5 ) + 5 ( 3 x − 5 ) = 540 5 x + 6 x + 10 x + 15 x + 180 = 540 36 x = 360 x = 10 Step 3: We can then find the measure of each interior angle: m ∡ A = 5 ( 10 + 7 ) = 85 ∘ m ∡ B = 120 ∘ m ∡ C = 6 ( 10 ) + 25 = 85 ∘ m ∡ D = 5 ( 2 * 10 + 5 ) = 125 ∘ m ∡ E = 5 ( 3 * 10 − 5 ) = 125 ∘ An exterior angle of a regular polygon is an angle formed by extending a side length beyond the closed figure. The measure of an exterior angle of a regular polygon with n sides is found using the following formula: To find the measure of an exterior angle of a regular polygon with n sides we use the formula b = 360 ∘ n . In , we have a regular hexagon ABCDEF . By extending the lines of each side, an angle is formed on the exterior of the hexagon at each vertex. The measure of each exterior angle is found using the formula, b = 360 ∘ 6 = 60 ∘ . Exterior Angles Now, an important point is that the sum of the exterior angles of a regular polygon with n sides equals 360 ∘ . This implies that when we multiply the measure of one exterior angle by the number of sides of the regular polygon, we should get 360 ∘ . For the example in , we multiply the measure of each exterior angle, 60 ∘ , by the number of sides, six. Thus, the sum of the exterior angles is 6 ( 60 ∘ ) = 360 ∘ . Calculating the Sum of Exterior Angles Find the sum of the measure of the exterior angles of the pentagon ( ). Each individual angle measures 360 5 = 72 ∘ . Then, the sum of the exterior angles is 5 ( 72 ∘ ) = 360 ∘ . Circles and Circumference The perimeter of a circle is called the circumference . To find the circumference, we use the formula C = π d , where d is the diameter, the distance across the center, or C = 2 π r , where r is the radius. The circumference of a circle is found using the formula C = π d , where d is the diameter of the circle, or C = 2 π r , where r is the radius. The radius is ½ of the diameter of a circle. The symbol π = 3.141592654 … is the ratio of the circumference to the diameter. Because this ratio is constant, our formula is accurate for any size circle. See . Circle Diameter and Radius Let the radius be equal to 3.5 inches. Then, the circumference is C = 2 π ( 3.5 ) = 21.99 in . Finding Circumference with Diameter Find the circumference of a circle with diameter 10 cm. If the diameter is 10 cm, the circumference is C = 10 π = 31.42 cm . Finding Circumference with Radius Find the radius of a circle with a circumference of 12 in. If the circumference is 12 in, then the radius is 12 = 2 π r 12 2 π = r = 1.91 in . Calculating Circumference for the Real World You decide to make a trim for the window in . How many feet of trim do you need to buy? The trim will cover the 6 feet along the bottom and the two 12-ft sides plus the half circle on top. The circumference of a semicircle is ½ the circumference of a circle. The diameter of the semicircle is 6 ft. Then, the circumference of the semicircle would be 1 2 π d = 1 2 π ( 6 ) = 3 π ft = 9.4 ft . Therefore, the total perimeter of the window is 6 + 12 + 12 + 9.4 = 39.4 ft . You need to buy 39.4 ft of trim. Archimedes Archimedes (credit: “Archimedes” Engraving from the book Les vrais pourtraits et vies des hommes illustres grecz, latins et payens (1586)/Wikimedia Commons, Public Domain) The overwhelming consensus is that Archimedes (287–212 BCE) was the greatest mathematician of classical antiquity, if not of all time. A Greek scientist, inventor, philosopher, astronomer, physicist, and mathematician, Archimedes flourished in Syracuse, Sicily. He is credited with the invention of various types of pulley systems and screw pumps based on the center of gravity. He advanced numerous mathematical concepts, including theorems for finding surface area and volume. Archimedes anticipated modern calculus and developed the idea of the “infinitely small” and the method of exhaustion. The method of exhaustion is a technique for finding the area of a shape inscribed within a sequence of polygons. The areas of the polygons converge to the area of the inscribed shape. This technique evolved to the concept of limits, which we use today. One of the more interesting achievements of Archimedes is the way he estimated the number pi, the ratio of the circumference of a circle to its diameter. He was the first to find a valid approximation. He started with a circle having a diameter of 1 inch. His method involved drawing a polygon inscribed inside this circle and a polygon circumscribed around this circle. He knew that the perimeter of the inscribed polygon was smaller than the circumference of the circle, and the perimeter of the circumscribed polygon was larger than the circumference of the circle. This is shown in the drawing of an eight-sided polygon. He increased the number of sides of the polygon each time as he got closer to the real value of pi. The following table is an example of how he did this. Sides Inscribed Perimeter Circumscribed Perimeter 4 2.8284 4.00 8 3.0615 3.3137 16 3.1214 3.1826 32 3.1365 3.1517 64 3.1403 3.1441 Archimedes settled on an approximation of π ≈ 3.1416 after an iteration of 96 sides. Because pi is an irrational number, it cannot be written exactly. However, the capability of the supercomputer can compute pi to billions of decimal digits. As of 2002, the most precise approximation of pi includes 1.2 trillion decimal digits. The Platonic Solids The Platonic solids ( ) have been known since antiquity. A polyhedron is a three-dimensional object constructed with congruent regular polygonal faces. Named for the philosopher, Plato believed that each one of the solids is associated with one of the four elements: Fire is associated with the tetrahedron or pyramid, earth with the cube, air with the octahedron, and water with the icosahedron. Of the fifth Platonic solid, the dodecahedron, Plato said, “… God used it for arranging the constellations on the whole heaven.” Platonic Solids Plato believed that the combination of these five polyhedra formed all matter in the universe. Later, Euclid proved that exactly five regular polyhedra exist and devoted the last book of the Elements to this theory. These ideas were resuscitated by Johannes Kepler about 2,000 years later. Kepler used the solids to explain the geometry of the universe. The beauty and symmetry of the Platonic solids have inspired architects and artists from antiquity to the present. Check Your Understanding Key Terms perimeter polygon pentagon hexagon heptagon octagon quadrilateral trapezoid parallelogram circumference Key Concepts Regular polygons are closed, two-dimensional figures that have equal side lengths. They are named for the number of their sides. The perimeter of a polygon is the measure of the outline of the shape. We determine a shape’s perimeter by calculating the sum of the lengths of its sides. The sum of the interior angles of a regular polygon with n sides is found using the formula S = ( n − 2 ) 180 ∘ . The measure of a single interior angle of a regular polygon with n sides is determined using the formula a = ( n − 2 ) 180 ∘ n . The sum of the exterior angles of a regular polygon is 360 ∘ . The measure of a single exterior angle of a regular polygon with n sides is found using the formula b = 360 ∘ n . The circumference of a circle is C = 2 π r , where r is the radius, or C = π d , and d is the diameter. Formulas The formula for the perimeter P of a rectangle is P = 2 L + 2 W , twice the length L plus twice the width W . The sum of the interior angles of a polygon with n sides is given by S = ( n − 2 ) 180 ∘ . The measure of each interior angle of a regular polygon with n sides is given by a = ( n − 2 ) 180 ∘ n . To find the measure of an exterior angle of a regular polygon with n sides we use the formula b = 360 ∘ n . The circumference of a circle is found using the formula C = π d , where d is the diameter of the circle, or C = 2 π r , where r is the radius.

Math