data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A circle is tangent to two adjacent sides \\(AB\\) and \\(AD\\) of square \\(ABCD\\) and cuts off a segment of length 8 cm from vertices \\(B\\) and \\(D\\) at the points of tangency. On the other two sides, the circle cuts off segments of 4 cm and 2 cm respectively from the ve... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1200,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle lies inside the parabola defined by the equation \\( y = 4x^2 \\) and is tangent to the parabola at two points. Determine how much higher the center of the circle is compared to the points of tangency."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 1201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle made of wire and a rectangle are arranged in such a way that the circle passes through two vertices $A$ and $B$ and touches the side $CD$. The length of side $CD$ is 32.1. Find the ratio of the sides of the rectangle, given that its perimeter is 4 times the radius of t... | math | {
"style": "rule",
"ground_truth": "4:1"
} | {
"index": 1202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle of diameter $1$ is removed from a $2\\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?\n[asy]\nfill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray);\ndraw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1));\nfill(circle((1,5/4),1/2),wh... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 1203,
"solution": "1. **Calculate the area of the rectangle**: \n The rectangle has dimensions $2 \\times 3$. Therefore, its area is:\n \\[\n \\text{Area}_{\\text{rectangle}} = 2 \\times 3 = 6.\n \\]\n\n2. **Determine the radius and area of the circle**:\n The diameter of the circle is given as... |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius $10$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. Calculate the measure of angle $ADB$."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 1204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius $10$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. The number of degrees of angle $ADB$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 1205,
"solution": "1. **Identify the properties of triangle $ABC$ and circle properties**: \n - Since $ABC$ is an equilateral triangle, each angle in the triangle is $60^\\circ$. Therefore, $\\angle ACB = 60^\\circ$.\n - The circle has its center at $C$ and passes through $A$ and $B$. Thus, $CA = CB ... |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius $15$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $BC$ extended through $C$ intersects the circle at point $E$. Find the number of degrees in angle $AEC$."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 1206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius $2$ has center at $(2,0)$. A circle of radius $1$ has center at $(5,0)$. A line is tangent to the two circles at points in the first quadrant. What is the $y$-intercept of the line?"
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 1207,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius $2$ is centered at $O$. Square $OABC$ has side length $1$. Sides $AB$ and $CB$ are extended past $B$ to meet the circle at $D$ and $E$, respectively. What is the area of the shaded region in the figure, which is bounded by $BD$, $BE$, and the minor arc connec... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}+1-\\sqrt{3}"
} | {
"index": 1208,
"solution": "1. **Identify the Geometry and Key Points**: We have a circle centered at $O$ with radius $2$, and a square $OABC$ with side length $1$. The sides $AB$ and $CB$ are extended to meet the circle at points $D$ and $E$ respectively.\n\n2. **Calculate $DA$ and $CE$ Using the Pythagorean The... |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius $2$ is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?\n[asy] size(0,50); draw((-1,1)..(-2,2)..(-3,1)..(-2,0)..cycle); dot((-1,1)); dot(... | math | {
"style": "rule",
"ground_truth": "\\frac{4-\\pi}{\\pi}"
} | {
"index": 1209,
"solution": "1. **Identify the Geometry of the Problem**: The problem states that a circle of radius $2$ is divided into four congruent arcs, which are then rearranged to form a star figure. The original circle has a radius of $2$, so its diameter is $4$.\n\n2. **Calculate the Area of the Original ... |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius $3$ is cut into six congruent arcs. These arcs are then rearranged symmetrically to form a hexagonal star as illustrated below. Determine the ratio of the area of the hexagonal star to the area of the original circle.\nA) $\\frac{4.5}{\\pi}$\nB) $\\frac{4.5\\... | math | {
"style": "rule",
"ground_truth": "\\frac{4.5\\sqrt{3}}{\\pi}"
} | {
"index": 1210,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius $r$ has chords $\\overline{AB}$ of length $10$ and $\\overline{CD}$ of length 7. When $\\overline{AB}$ and $\\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at $P$, which is outside of the circle. If $\\angle{APD}=60^\\circ$ and ... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 1211,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius $r$ has chords $\\overline{AB}$ of length $12$ and $\\overline{CD}$ of length $9$. When $\\overline{AB}$ and $\\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at point $P$, which is outside of the circle. If $\\angle{APD}=90^\\ci... | math | {
"style": "rule",
"ground_truth": "221"
} | {
"index": 1212,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius $r$ is concentric with and outside a regular hexagon of side length $2$. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is $1/2$. What is $r$?"
}
] | math | {
"style": "rule",
"ground_truth": "$3\\sqrt{2}+\\sqrt{6}$"
} | {
"index": 1213,
"solution": "1. **Understanding the Problem**: We are given a regular hexagon with side length $2$ and a concentric circle with radius $r$. We need to find $r$ such that the probability of seeing exactly three sides of the hexagon from a random point on the circle is $1/2$.\n\n2. **Visualizing the ... |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16.$ The set of all possible values of $r$ is an interval $[a,b).$ What is $a+b?$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{15}+8"
} | {
"index": 1214,
"solution": "To solve this problem, we first need to understand the properties of the ellipse and the circle described in the problem.\n\n1. **Identify the ellipse properties:**\n The given ellipse equation is \\(x^2 + 16y^2 = 16\\). We can rewrite this equation in standard form:\n \\[\n \\fr... |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius 1 is internally tangent to two circles of radius 2 at points $A$ and $B$, where $AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the figure, that is outside the smaller circle and inside each of the two larger circles? Ex... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}\\pi - 2\\sqrt{3}"
} | {
"index": 1215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + ... | math | {
"style": "rule",
"ground_truth": "817"
} | {
"index": 1216,
"solution": "Let the bisector of $\\angle CAD$ be $AE$, with $E$ on $CD$. By the angle bisector theorem, $DE = 36/5$. Since $\\triangle AOR \\sim \\triangle AED$ ($O$ is the center of the circle), we find that $AR = 5$ since $OR = 1$. Also $AT = 35$ so $RT = OQ = 30$.\n We can apply the same princi... |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius 1 is surrounded by 4 circles of radius $r$ as shown. What is $r$?\n\n[asy]\nunitsize(0.6cm);\nfor(int i=0; i<2; ++i){\nfor(int j=0; j<2; ++j){\ndraw(Circle((-2.4+4.8i,-2.4+4.8j),2.4),linewidth(0.7));\ndraw((-2.4+4.8i,-2.4+4.8j)--(-0.7+4.8i,-0.7+4.8j));\nlabel... | math | {
"style": "rule",
"ground_truth": "1+\\sqrt{2}"
} | {
"index": 1217,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius 1 is tangent to a circle of radius 2. The sides of $\\triangle ABC$ are tangent to the circles as shown, and the sides $\\overline{AB}$ and $\\overline{AC}$ are congruent. What is the area of $\\triangle ABC$?\n\n[asy]\nunitsize(0.7cm);\npair A,B,C;\nA=(0,8);... | math | {
"style": "rule",
"ground_truth": "16\\sqrt{2}"
} | {
"index": 1218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius 2 is centered at $A$. An equilateral triangle with side 4 has a vertex at $A$. What is the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the ... | math | {
"style": "rule",
"ground_truth": "$4(\\pi - \\sqrt{3})$"
} | {
"index": 1219,
"solution": "1. **Calculate the area of the circle:**\n The formula for the area of a circle is $\\pi r^2$. Given the radius $r = 2$, the area of the circle is:\n \\[\n \\text{Area of the circle} = \\pi \\times 2^2 = 4\\pi\n \\]\n\n2. **Calculate the area of the equilateral triangle:**\n ... |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius 3 is centered at point $A$. An equilateral triangle with side length 6 has one vertex tangent to the edge of the circle at point $A$. Calculate the difference between the area of the region that lies inside the circle but outside the triangle and the area of ... | math | {
"style": "rule",
"ground_truth": "9(\\sqrt{3} - \\pi)"
} | {
"index": 1220,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 1221,
"solution": "1. **Identify the relationship between the circle and the rectangle**: The circle is inscribed in the rectangle, which means the diameter of the circle is equal to the height (shorter side) of the rectangle.\n\n2. **Calculate the diameter of the circle**: Given that the radius of the c... |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius 6 is drawn centered at the origin. How many squares of side length 1 and integer coordinate vertices intersect the interior of this circle?"
}
] | math | {
"style": "rule",
"ground_truth": "132"
} | {
"index": 1222,
"solution": "By symmetry, the answer is four times the number of squares in the first quadrant. Let's identify each square by its coordinates at the bottom-left corner, $(x, y)$. When $x=0$, we can have $y=0 \\ldots 5$, so there are 6 squares. (Letting $y=6$ is not allowed because that square inter... |
DeepScaleR | [
{
"role": "user",
"content": "A circle of radius \\( 2 \\) cm is inscribed in \\( \\triangle ABC \\). Let \\( D \\) and \\( E \\) be the points of tangency of the circle with the sides \\( AC \\) and \\( AB \\), respectively. If \\( \\angle BAC = 45^{\\circ} \\), find the length of the minor arc \\( DE \\).... | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"index": 1223,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle on a plane divides the plane into 2 parts. A circle and a line can divide the plane into a maximum of 4 parts. A circle and 2 lines can divide the plane into a maximum of 8 parts. A circle and 5 lines can divide the plane into a maximum of how many parts?"
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 1224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle passes through the midpoints of the hypotenuse $AB$ and the leg $BC$ of the right triangle $ABC$ and touches the leg $AC$. In what ratio does the point of tangency divide the leg $AC$?"
}
] | math | {
"style": "rule",
"ground_truth": "1 : 3"
} | {
"index": 1225,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle passes through the point $(0,1),$ and is tangent to the parabola $y = x^2$ at $(2,4).$ Find the center of the circle.\n\n[asy]\nunitsize(0.4 cm);\n\nreal parab (real x) {\n return(x^2);\n}\n\ndraw(graph(parab,-3.5,3.5));\ndraw(Circle((-16/5,53/10),13*sqrt(17)/10));\n... | math | {
"style": "rule",
"ground_truth": "\\left( -\\frac{16}{5}, \\frac{53}{10} \\right)"
} | {
"index": 1226,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle passes through the points $(2,0)$ and $(4,0)$ and is tangent to the line $y=x$. Find the sum of all possible values for the $y$-coordinate of the center of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 1227,
"solution": "First, we see that the x coordinate must be 3. Let the y coordinate be y. Now, we see that the radius is $r=\\sqrt{1+y^{2}}$. The line from the center of the circle to the point of tangency with the line $x=y$ is perpendicular to the line $x=y$. Hence, the distance from the center of t... |
DeepScaleR | [
{
"role": "user",
"content": "A circle passes through the three vertices of an isosceles triangle that has two sides of length 3 and a base of length 2. What is the area of this circle? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{81}{32}\\pi"
} | {
"index": 1228,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle passes through the three vertices of an isosceles triangle that has two sides of length 5 and a base of length 4. What is the area of this circle? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13125}{1764}\\pi"
} | {
"index": 1229,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle passes through the vertex of a rectangle $ABCD$ and touches its sides $AB$ and $AD$ at $M$ and $N$ respectively. If the distance from $C$ to the line segment $MN$ is equal to $5$ units, find the area of rectangle $ABCD$ ."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 1230,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle passes through the vertices $K$ and $P$ of triangle $KPM$ and intersects its sides $KM$ and $PM$ at points $F$ and $B$, respectively. Given that $K F : F M = 3 : 1$ and $P B : B M = 6 : 5$, find $K P$ given that $B F = \\sqrt{15}$."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{33}"
} | {
"index": 1231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle passes through the vertices of a triangle with side-lengths $7\\tfrac{1}{2},10,12\\tfrac{1}{2}.$ The radius of the circle is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{4}"
} | {
"index": 1232,
"solution": "1. **Identify the Type of Triangle**: \n Given the side lengths of the triangle are $7\\tfrac{1}{2}$, $10$, and $12\\tfrac{1}{2}$. We first convert these mixed fractions to improper fractions for easier calculation:\n \\[\n 7\\tfrac{1}{2} = \\frac{15}{2}, \\quad 12\\tfrac{1}{2} ... |
DeepScaleR | [
{
"role": "user",
"content": "A circle passes through vertex $B$ of the triangle $ABC$, intersects its sides $ AB $and $BC$ at points $K$ and $L$, respectively, and touches the side $ AC$ at its midpoint $M$. The point $N$ on the arc $BL$ (which does not contain $K$) is such that $\\angle LKN = \\angle ACB$... | math | {
"style": "rule",
"ground_truth": "75^\\circ"
} | {
"index": 1233,
"solution": "\nWe are given a triangle \\( ABC \\) with a circle that touches the side \\( AC \\) at its midpoint \\( M \\), passes through the vertex \\( B \\), and intersects \\( AB \\) and \\( BC \\) at \\( K \\) and \\( L \\), respectively. The point \\( N \\) is located on the arc \\( BL \\) (... |
DeepScaleR | [
{
"role": "user",
"content": "A circle passing through the vertex \\( P \\) of triangle \\( PQR \\) touches side \\( QR \\) at point \\( F \\) and intersects sides \\( PQ \\) and \\( PR \\) at points \\( M \\) and \\( N \\), respectively, different from vertex \\( P \\). Find the ratio \\( QF : FR \\) if it... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 1234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle rests in the interior of the parabola with equation $y = x^2,$ so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 1235,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$ with a side length of $2-\\sqrt{5-\\sqrt{5}}$ cm. From point $C$, two tangents are drawn to this circle. Find the radius of the circle, given that the angle between the tangents is $72^{\\circ}$ and it ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5 - \\sqrt{5}}"
} | {
"index": 1236,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle touches the extensions of two sides \\( AB \\) and \\( AD \\) of a square \\( ABCD \\) with a side length of 4 cm. From point \\( C \\), two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is \\( 60^{\\circ} \\)."
}
... | math | {
"style": "rule",
"ground_truth": "4 (\\sqrt{2} + 1)"
} | {
"index": 1237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle touches the extensions of two sides \\(AB\\) and \\(AD\\) of square \\(ABCD\\), and the point of tangency cuts off a segment of length \\(2 + \\sqrt{5 - \\sqrt{5}}\\) cm from vertex \\(A\\). From point \\(C\\), two tangents are drawn to this circle. Find the side lengt... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{\\sqrt{5} - 1} \\cdot \\sqrt[4]{125}}{5}"
} | {
"index": 1238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle touches the longer leg of a right triangle, passes through the vertex of the opposite acute angle, and has its center on the hypotenuse of the triangle. What is the radius of the circle if the lengths of the legs are 5 and 12?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{65}{18}"
} | {
"index": 1239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle touches the sides of an angle at points \\( A \\) and \\( B \\). The distance from a point \\( C \\) lying on the circle to the line \\( AB \\) is 6. Find the sum of the distances from point \\( C \\) to the sides of the angle, given that one of these distances is nine... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 1240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a circumscribed and an inscribed square centered at the origin of a rectangular coordinate system with positive $x$ and $y$ axes is shown in each figure I to IV below.\n\nThe inequalities\n\\(|x|+|y| \\leq \\sqrt{2(x^{2}+y^{2})} \\leq 2\\mbox{Max}(|x|, |y|)\\)\nar... | math | {
"style": "rule",
"ground_truth": "II"
} | {
"index": 1241,
"solution": "To solve this problem, we need to analyze the given inequalities and understand their geometric representations. We will consider each inequality separately and then find the figure that correctly represents all three inequalities simultaneously.\n\n1. **Understanding the Inequalities:... |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 15 is tangent to two adjacent sides \\( AB \\) and \\( AD \\) of square \\( ABCD \\). On the other two sides, the circle intercepts segments of 6 and 3 cm from the vertices, respectively. Find the length of the segment that the circle intercepts from v... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 1242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 2 is inscribed in triangle \\(ABC\\) and touches side \\(BC\\) at point \\(D\\). Another circle with a radius of 4 touches the extensions of sides \\(AB\\) and \\(AC\\), as well as side \\(BC\\) at point \\(E\\). Find the area of triangle \\(ABC\\) if ... | math | {
"style": "rule",
"ground_truth": "\\frac{56}{\\sqrt{3}}"
} | {
"index": 1243,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 2 passes through the midpoints of three sides of triangle \\(ABC\\), where the angles at vertices \\(A\\) and \\(B\\) are \\(30^{\\circ}\\) and \\(45^{\\circ}\\), respectively.\n\nFind the height drawn from vertex \\(A\\)."
}
] | math | {
"style": "rule",
"ground_truth": "2 + 2\\sqrt{3}"
} | {
"index": 1244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 2 units has its center at $(0, 0)$. A circle with a radius of 7 units has its center at $(15, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common ... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"index": 1245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 2 units rolls around the inside of a triangle with sides 9, 12, and 15 units. The circle is always tangent to at least one side of the triangle. Calculate the total distance traveled by the center of the circle when it returns to its starting position.... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 1246,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 3 is centered at the midpoint of one side of an equilateral triangle each side of which has a length of 9. Determine the difference between the area inside the circle but outside the triangle and the area inside the triangle but outside the circle."
... | math | {
"style": "rule",
"ground_truth": "9\\pi - \\frac{81\\sqrt{3}}{4}"
} | {
"index": 1247,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 3 is inscribed in a right trapezoid, where the shorter base is 4. Find the length of the longer base of the trapezoid."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 1248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 1249,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 8 units has its center at $(18, 0)$. A line tangent to both circles intersects the $x$-axis at point $(x, 0)$ to the right of the origin. Find the value of $x$. Express your answer as a comm... | math | {
"style": "rule",
"ground_truth": "\\frac{54}{11}"
} | {
"index": 1250,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 1251,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both these circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answe... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 1252,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 5 units has its center at $(12, 0)$. Find the x-coordinate of the point on the $x$-axis where a line, tangent to both circles, intersects. The line should intersect the x-axis to the r... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"index": 1253,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 3 units has its center at $(0, 0)$. Another circle with a radius of 8 units has its center at $(20, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. Determine the value of $x$. Express your answer as a... | math | {
"style": "rule",
"ground_truth": "\\frac{60}{11}"
} | {
"index": 1254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 6 cm is tangent to three sides of a rectangle. The area of the circle is half the area of the rectangle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "6\\pi"
} | {
"index": 1255,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with a radius of 6 is inscribed around the trapezoid \\(ABCD\\). The center of this circle lies on the base \\(AD\\), and \\(BC = 4\\). Find the area of the trapezoid."
}
] | math | {
"style": "rule",
"ground_truth": "24\\sqrt{2}"
} | {
"index": 1256,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with area $A_1$ is contained in the interior of a larger circle with area $A_1+A_2$. If the radius of the larger circle is $3$, and if $A_1 , A_2, A_1 + A_2$ is an arithmetic progression, then the radius of the smaller circle is"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 1257,
"solution": "1. **Identify the areas of the circles**: \n Given that the area of the larger circle is $A_1 + A_2$ and its radius is $3$, we can calculate its area using the formula for the area of a circle, $A = \\pi r^2$. Thus, the area of the larger circle is:\n \\[\n A_1 + A_2 = \\pi \\tim... |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center $O$ has radius $5,$ and has two points $A,B$ on the circle such that $\\angle AOB = 90^{\\circ}.$ Rays $OA$ and $OB$ are extended to points $C$ and $D,$ respectively, such that $AB$ is parallel to $CD,$ and the length of $CD$ is $200\... | math | {
"style": "rule",
"ground_truth": "5 + \\frac{15\\sqrt{2}}{2}"
} | {
"index": 1258,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center $A$ and radius three inches is tangent at $C$ to a circle with center $B$, as shown. If point $B$ is on the small circle, what is the area of the shaded region? Express your answer in terms of $\\pi$.\n\n[asy]\nfilldraw(circle((0,0),6),gray,linewidth(2));\n... | math | {
"style": "rule",
"ground_truth": "27\\pi"
} | {
"index": 1259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center $A$ has radius $10$ units and circle $B$ has radius $3$ units. The circles are externally tangent to each other at point $C$. Segment $XY$ is the common external tangent to circle $A$ and circle $B$ at points $X$ and $Y$, respectively. What is the length of... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{55}"
} | {
"index": 1260,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center $C$ is tangent to the positive $x$ and $y$-axes and externally tangent to the circle centered at $(3,0)$ with radius $1$. What is the sum of all possible radii of the circle with center $C$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 1261,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center $D$ and radius four feet is tangent at $E$ to a circle with center $F$, as shown. If point $F$ is on the small circle, what is the area of the shaded region? Express your answer in terms of $\\pi$.\n\n[asy]\nfilldraw(circle((0,0),8),gray,linewidth(2));\nfil... | math | {
"style": "rule",
"ground_truth": "48\\pi"
} | {
"index": 1262,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center $O$ and equation $x^2 + y^2 = 1$ passes through point $P(-1, \\sqrt{3})$. Two tangents are drawn from $P$ to the circle, touching the circle at points $A$ and $B$ respectively. Find the length of the chord $|AB|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 1263,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center $O$ has area $156\\pi$. Triangle $ABC$ is equilateral, $\\overline{BC}$ is a chord on the circle, $OA = 4\\sqrt{3}$, and point $O$ is outside $\\triangle ABC$. What is the side length of $\\triangle ABC$?"
}
] | math | {
"style": "rule",
"ground_truth": "$6$"
} | {
"index": 1264,
"solution": "1. **Calculate the radius of the circle**:\n The area of the circle is given by the formula $A = \\pi r^2$. Given $A = 156\\pi$, we solve for $r$:\n \\[\n \\pi r^2 = 156\\pi \\implies r^2 = 156 \\implies r = \\sqrt{156} = 12\\sqrt{3}\n \\]\n\n2. **Analyze the position of point ... |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center $O$ has radius $8$ units and circle $P$ has radius $2$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. What is the length ... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2}"
} | {
"index": 1265,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center $O$ has radius 25. Chord $\\overline{AB}$ of length 30 and chord $\\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\\frac{m}{n}$, where $m$ and $n$ ar... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 1266,
"solution": "Let $OP=x$.\n Proceed as the first solution in finding that quadrilateral $EPFO$ has side lengths $OE=20$, $OF=24$, $EP=\\sqrt{x^2-20^2}$, and $PF=\\sqrt{x^2-24^2}$, and diagonals $OP=x$ and $EF=12$.\n We note that quadrilateral $EPFO$ is cyclic and use Ptolemy's theorem to solve for $... |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\\circ$-$60^\\circ$-$90^\\circ$ triangle $ABC$ as shown, where $AB=1$. To the nearest hundredth, what is the radius of the circle?"
}
] | math | {
"style": "rule",
"ground_truth": "2.37"
} | {
"index": 1267,
"solution": "1. **Identify the Geometry and Setup**: We are given a $30^\\circ$-$60^\\circ$-$90^\\circ$ triangle $ABC$ with $AB=1$. The circle with center $O$ is tangent to the coordinate axes and the hypotenuse of triangle $ABC$. We need to find the radius $r$ of this circle.\n\n2. **Label the Poi... |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center O is tangent to the coordinate axes and to the hypotenuse of a $45^\\circ$-$45^\\circ$-$90^\\circ$ triangle ABC, where AB = 2. Determine the exact radius of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{2}"
} | {
"index": 1268,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center P and radius 4 inches is tangent at D to a circle with center Q, located at a 45-degree angle from P. If point Q is on the smaller circle, what is the area of the shaded region? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "48\\pi"
} | {
"index": 1269,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center \\( Q \\) and radius 2 rolls around the inside of a right triangle \\( DEF \\) with side lengths 9, 12, and 15, always remaining tangent to at least one side of the triangle. When \\( Q \\) first returns to its original position, through what distance has \... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 1270,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with center at $O$ has radius 1. Points $P$ and $Q$ outside the circle are placed such that $P Q$ passes through $O$. Tangent lines to the circle through $P$ hit the circle at $P_{1}$ and $P_{2}$, and tangent lines to the circle through $Q$ hit the circle at $Q_{1}$ an... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{12}"
} | {
"index": 1271,
"solution": "$(45-30)^{\\circ}=\\frac{\\pi}{12}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with diameter $AB$ is drawn, and the point $ P$ is chosen on segment $AB$ so that $\\frac{AP}{AB} =\\frac{1}{42}$ . Two new circles $a$ and $b$ are drawn with diameters $AP$ and $PB$ respectively. The perpendicular line to $AB$ passing through $ P$ ... | math | {
"style": "rule",
"ground_truth": "205/1681"
} | {
"index": 1272,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with diameter $\\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$. The leng... | math | {
"style": "rule",
"ground_truth": "312"
} | {
"index": 1273,
"solution": "Call the center of the larger circle $O$. Extend the diameter $\\overline{PQ}$ to the other side of the square (at point $E$), and draw $\\overline{AO}$. We now have a right triangle, with hypotenuse of length $20$. Since $OQ = OP - PQ = 20 - 10 = 10$, we know that $OE = AB - OQ = AB -... |
DeepScaleR | [
{
"role": "user",
"content": "A circle with equation $x^{2}+y^{2}=1$ passes through point $P(1, \\sqrt {3})$. Two tangents are drawn from $P$ to the circle, touching the circle at points $A$ and $B$ respectively. Find the length of the chord $|AB|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"index": 1274,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with its center at point $M$ on the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) is tangent to the $x$-axis exactly at one of the foci $F$ of the hyperbola, and intersects the $y$-axis at points $P$ and $Q$. If $\\triangle MPQ$ is an equilateral... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 1275,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with its center on side $AB$ of triangle $ABC$ touches the other two sides. Find the area of the circle if $a = 13$ cm, $b = 14$ cm, and $c = 15$ cm, where $a$, $b$, and $c$ are the lengths of the sides of the triangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3136}{81} \\pi"
} | {
"index": 1276,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with its center on the line \\( y = b \\) intersects the parabola \\( y = \\frac{3}{4} x^{2} \\) at least at three points; one of these points is the origin, and two of the remaining points lie on the line \\( y = \\frac{3}{4} x + b \\). Find all values of \\( b \\) fo... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{12}"
} | {
"index": 1277,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with radius $6$ is externally tangent to a circle with radius $24$. Find the area of the triangular region bounded by the three common tangent lines of these two circles."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 1278,
"solution": "[asy] //Created by isabelchen size(12cm, 12cm); draw(circle((0,0),24)); draw(circle((30,0),6)); draw((72/5, 96/5) -- (40,0)); draw((72/5, -96/5) -- (40,0)); draw((24, 12) -- (24, -12)); draw((0, 0) -- (40, 0)); draw((72/5, 96/5) -- (0,0)); draw((168/5, 24/5) -- (30,0)); draw((54/5, 72/... |
DeepScaleR | [
{
"role": "user",
"content": "A circle with radius $\\frac{\\sqrt{2}}{2}$ and a regular hexagon with side length 1 share the same center. Calculate the area inside the circle, but outside the hexagon."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2} - \\frac{3\\sqrt{3}}{2}"
} | {
"index": 1279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with radius $r$ is contained within the region bounded by a circle with radius $R$. The area bounded by the larger circle is $\\frac{a}{b}$ times the area of the region outside the smaller circle and inside the larger circle. Then $R:r$ equals:"
}
] | math | {
"style": "rule",
"ground_truth": "$\\sqrt{a}:\\sqrt{a-b}$"
} | {
"index": 1280,
"solution": "1. **Identify the areas involved**: \n - The area of the larger circle is $\\pi R^2$.\n - The area of the smaller circle is $\\pi r^2$.\n - The area of the region outside the smaller circle and inside the larger circle is $\\pi R^2 - \\pi r^2$.\n\n2. **Set up the equation based o... |
DeepScaleR | [
{
"role": "user",
"content": "A circle with radius $r$ is tangent to sides $AB, AD$ and $CD$ of rectangle $ABCD$ and passes through the midpoint of diagonal $AC$. The area of the rectangle, in terms of $r$, is"
}
] | math | {
"style": "rule",
"ground_truth": "$8r^2$"
} | {
"index": 1281,
"solution": "Let's analyze the problem and derive the solution step-by-step.\n\n1. **Understanding the Geometry**:\n - The circle is tangent to sides $AB$, $AD$, and $CD$ of rectangle $ABCD$. This implies that the radius $r$ of the circle is the distance from the center of the circle to each of t... |
DeepScaleR | [
{
"role": "user",
"content": "A circle with radius 1 is tangent to a circle with radius 3 at point \\( C \\). A line passing through point \\( C \\) intersects the smaller circle at point \\( A \\) and the larger circle at point \\( B \\). Find \\( AC \\), given that \\( AB = 2\\sqrt{5} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"index": 1282,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with radius 100 is drawn on squared paper with unit squares. It does not touch any of the grid lines or pass through any of the lattice points. What is the maximum number of squares it can pass through?"
}
] | math | {
"style": "rule",
"ground_truth": "800"
} | {
"index": 1283,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with radius 4 cm is tangent to three sides of a rectangle, as shown. The area of the rectangle is twice the area of the circle. What is the length of the longer side of the rectangle, in centimeters? Express your answer in terms of $\\pi$.\n\n[asy]\nimport graph;\ndraw... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 1284,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with radius 6 cm is tangent to three sides of a rectangle. The area of the rectangle is three times the area of the circle. Determine the length of the longer side of the rectangle, expressed in centimeters and in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "9\\pi"
} | {
"index": 1285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle with radius \\(5\\) has its center on the \\(x\\)-axis, and the \\(x\\)-coordinate of the center is an integer. The circle is tangent to the line \\(4x+3y-29=0\\).\n(Ⅰ) Find the equation of the circle;\n(Ⅱ) Let the line \\(ax-y+5=0\\) (\\(a > 0\\)) intersect the circle... | math | {
"style": "rule",
"ground_truth": "a = \\dfrac {3}{4}"
} | {
"index": 1286,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circle, whose center lies on the line \\( y = b \\), intersects the parabola \\( y = \\frac{12}{5} x^2 \\) at least at three points; one of these points is the origin, and two of the remaining points lie on the line \\( y = \\frac{12}{5} x + b \\). Find all values of \\( b \\... | math | {
"style": "rule",
"ground_truth": "169/60"
} | {
"index": 1287,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circular cylindrical post has a circumference of 6 feet and a height of 18 feet. A string is wrapped around the post which spirals evenly from the bottom to the top, looping around the post exactly six times. What is the length of the string, in feet?"
}
] | math | {
"style": "rule",
"ground_truth": "18\\sqrt{5}"
} | {
"index": 1288,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circular cylindrical post with a circumference of 4 feet has a string wrapped around it, spiraling from the bottom of the post to the top of the post. The string evenly loops around the post exactly four full times, starting at the bottom edge and finishing at the top edge. T... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 1289,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circular dartboard is divided into regions with various central angles, as shown. The probability of a dart randomly landing in a particular region is $\\frac16$. What is the corresponding measure, in degrees, of the central angle of this section of the dartboard? [asy]\nunit... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 1290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circular disc with diameter $D$ is placed on an $8 \\times 8$ checkerboard with width $D$ so that the centers coincide. The number of checkerboard squares which are completely covered by the disc is"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 1291,
"solution": "1. **Understanding the Problem**: We are given a circular disc with diameter $D$ placed on an $8 \\times 8$ checkerboard such that the centers of both the disc and the checkerboard coincide. We need to find the number of squares completely covered by the disc.\n\n2. **Checkerboard and ... |
DeepScaleR | [
{
"role": "user",
"content": "A circular disk is divided by $2n$ equally spaced radii ($n>0$) and one secant line. The maximum number of non-overlapping areas into which the disk can be divided is"
}
] | math | {
"style": "rule",
"ground_truth": "3n+1"
} | {
"index": 1292,
"solution": "To solve this problem, we need to determine how the addition of $2n$ radii and one secant line divides a circular disk into non-overlapping areas. We will analyze the pattern by considering the effect of each additional radius and the secant line.\n\n1. **Base Case Analysis**:\n - Wh... |
DeepScaleR | [
{
"role": "user",
"content": "A circular garden is enlarged so that the new diameter is twice the old diameter. What is the ratio of the original area to the enlarged area? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 1293,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circular grass plot 12 feet in diameter is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is"
}
] | math | {
"style": "rule",
"ground_truth": "30\\pi - 9\\sqrt3"
} | {
"index": 1294,
"solution": "1. **Identify the dimensions of the plot and path**: The circular grass plot has a diameter of 12 feet, so its radius \\( r \\) is \\( \\frac{12}{2} = 6 \\) feet. The gravel path is 3 feet wide.\n\n2. **Calculate the area of the entire circle**: The area \\( A \\) of a circle is given ... |
DeepScaleR | [
{
"role": "user",
"content": "A circular paper with a radius of 6 inches has a section removed to form a \\(240^\\circ\\) sector. This sector is then used to form a right circular cone by bringing the two radii together. Find the circumference of the base of the cone in terms of \\(\\pi\\)."
}
] | math | {
"style": "rule",
"ground_truth": "8\\pi"
} | {
"index": 1295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circular piece of metal with a radius of 10 units has a maximum square cut out from it. Then, the largest possible circle is cut from this square. Calculate the total amount of metal wasted.\nA) $50\\pi - 200$\nB) $200 - 50\\pi$\nC) $100\\pi$\nD) $50\\pi$\nE) None of these"
... | math | {
"style": "rule",
"ground_truth": "50\\pi - 200"
} | {
"index": 1296,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circular sheet of iron with a radius of 6 has a sector removed, which is $\\frac{1}{6}$ of the original area. The remaining part is rolled into the lateral surface of a cone. The volume of the cone is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25\\sqrt{11}}{3}\\pi"
} | {
"index": 1297,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circular sheet of paper with a radius of $12$ cm is cut into four congruent sectors. A cone is formed by rolling one of these sectors until the edges meet. Calculate both the height and the volume of this cone. Express the height in simplest radical form and the volume in ter... | math | {
"style": "rule",
"ground_truth": "9\\pi\\sqrt{15}"
} | {
"index": 1298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circular sheet of paper with radius of $6$ cm is cut into three congruent sectors. What is the height of the cone in centimeters that can be created by rolling one of the sections until the edges meet? Express your answer in simplest radical form."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{2}"
} | {
"index": 1299,
"solution": ""
} |
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