Yuting6/geoqa-r1v-8k-rotated huggingface dataset:
Problem: <image>Question: The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?
<image1>
Answer: 4
Problem: <image>Question: Rectangle $DEFA$ below is a $3 \times 4$ rectangle with $DC=CB=BA$.  The area of the "bat wings" is
<image1>
Choices:
A. $2$
B. $2 \frac{1}{2}$
C. $3$
D. $3 \frac{1}{2}$
E. $5$
Answer: C
Problem: <image>Question: A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?

<image1>
Choices:
A. $4 \sqrt{3}$
B. $\frac{120}{17}$
C. $10$
D. $\frac{17\sqrt{2}}{2}$
E. $\frac{17\sqrt{3}}{2}$
Answer: B
Problem: <image>Question: Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received 36 votes, then how many votes were cast all together?

<image1>
Answer: 120
Problem: <image>Question: In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path allows only moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture.
<image1>
Answer: 24
Problem: <image>Question: In the figure below, choose point $D$ on $\overline{BC}$ so that $\triangle ACD$ and $\triangle ABD$ have equal perimeters. What is the area of $\triangle ABD$?
<image1>
Choices:
A. $\frac{3}{4}$
B. $\frac{3}{2}$
C. $2$
D. $\frac{12}{5}$
E. $\frac{5}{2}$
Answer: D
Problem: <image>Question: In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$.
<image1>
What is the area of quadrilateral $ABCD$?
Answer: 24
Problem: <image>Question: In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?
<image1>
Choices:
A. $\frac{7}{6}$
B. $\frac{13}{5}$
C. $\frac{59}{18}$
D. $\frac{10}{3}$
E. $\frac{60}{13}$
Answer: D
Problem: <image>Question: In the figure shown, $\overline{US}$ and $\overline{UT}$ are line segments each of length 2, and $m\angle TUS = 60^\circ$. Arcs $\overarc{TR}$ and $\overarc{SR}$ are each one-sixth of a circle with radius 2. What is the area of the region shown?
<image1>
Choices:
A. $3\sqrt{3}-\pi$
B. $4\sqrt{3}-\frac{4\pi}{3}$
C. $2\sqrt{3}$
D. $4\sqrt{3}-\frac{2\pi}{3}$
E. $4+\frac{4\pi}{3}$
Answer: B
Problem: <image>Question: The twelve-sided figure shown has been drawn on $1 \text{ cm}\times 1 \text{ cm}$ graph paper. What is the area of the figure in $\text{cm}^2$?

<image1>
Answer: 13
Problem: <image>Question: Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.

<image1>What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class?
Answer: 4.36
Problem: <image>Question: In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units?

<image1>
Choices:
A. $\frac{1}{4}$
B. $\frac{1}{3}$
C. $\frac{1}{2}$
D. $1$
E. $\frac{\pi}{2}$
Answer: D
Problem: <image>Question: In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid?

<image1>
Answer: 8
Problem: <image>Question: In $\triangle ABC,$ a point $E$ is on $\overline{AB}$ with $AE=1$ and $EB=2$. Point $D$ is on $\overline{AC}$ so that $\overline{DE} \parallel \overline{BC}$ and point $F$ is on $\overline{BC}$ so that $\overline{EF} \parallel \overline{AC}$. What is the ratio of the area of $CDEF$ to the area of $\triangle ABC?$

<image1>
Choices:
A. $\frac{4}{9}$
B. $\frac{1}{2}$
C. $\frac{5}{9}$
D. $\frac{3}{5}$
E. $\frac{2}{3}$
Answer: A
Problem: <image>Question: Point $E$ is the midpoint of side $\overline{CD}$ in square $ABCD,$ and $\overline{BE}$ meets diagonal $\overline{AC}$ at $F$. The area of quadrilateral $AFED$ is $45$. What is the area of $ABCD?$

<image1>
Answer: 108
Problem: <image>Question: From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?

<image1>
Choices:
A. $\frac{2}{7}$
B. $\frac{5}{42}$
C. $\frac{11}{14}$
D. $\frac{5}{7}$
E. $\frac{6}{7}$
Answer: D
Problem: <image>Question: In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of edges $\overline{FB}$ and $\overline{HD},$ respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2?$

<image1>
Choices:
A. $\frac{5}{4}$
B. $\frac{4}{3}$
C. $\frac{3}{2}$
D. $\frac{25}{16}$
E. $\frac{9}{4}$
Answer: C
Problem: <image>Question: Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles $5$ feet, what is the area in square feet of rectangle $ABCD$?
<image1>
Answer: 150
Problem: <image>Question: Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$?
<image1>
Answer: 120
Problem: <image>Question: A tortoise challenges a hare to a race. The hare eagerly agrees and quickly runs ahead, leaving the slow-moving tortoise behind. Confident that he will win, the hare stops to take a nap. Meanwhile, the tortoise walks at a slow steady pace for the entire race. The hare awakes and runs to the finish line, only to find the tortoise already there. Which of the following graphs matches the description of the race, showing the distance $d$ traveled by the two animals over time $t$ from start to finish?$\phantom{h}$

<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: B
Problem: <image>Question: There are 81 grid points (uniformly spaced) in the square shown in the diagram below, including the points on the edges. Point $P$ is the center of the square. Given that point $Q$ is randomly chosen from among the other 80 points, what is the probability that line $PQ$ is a line of symmetry for the square?
<image1>
Choices:
A. $\frac{1}{5}$
B. $\frac{1}{4}$
C. $\frac{2}{5}$
D. $\frac{9}{20}$
E. $\frac{1}{2}$
Answer: C
Problem: <image>Question: The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually $21$ participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?

<image1>
Choices:
A. $\text{The mean increases by 1 and the median does not change.}$
B. $\text{The mean increases by 1 and the median increases by 1.}$
C. $\text{The mean increases by 1 and the median increases by 5.}$
D. $\text{The mean increases by 5 and the median increases by 1.}$
E. $\text{The mean increases by 5 and the median increases by 5.}$
Answer: B
Problem: <image>Question: The faces of a cube are painted in six different colors: red (R), white (W), green (G), brown (B), aqua (A), and purple (P). Three views of the cube are shown below. What is the color of the face opposite the aqua face?
<image1>
Choices:
A. $\text{red}$
B. $\text{white}$
C. $\text{green}$
D. $\text{brown}$
E. $\text{purple}
$
Answer: A
Problem: <image>Question: In triangle $ABC$, point $D$ divides side $\overline{AC}$ so that $AD:DC=1:2$. Let $E$ be the midpoint of $\overline{BD}$ and let $F$ be the point of intersection of line $BC$ and line $AE$. Given that the area of $\triangle ABC$ is $360$, what is the area of $\triangle EBF$?
<image1>
Answer: 30
Problem: <image>Question: Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?

<image1>
Answer: 37
Problem: <image>Question: Akash's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides?

<image1>
Answer: 20
Problem: <image>Question: After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?

<image1>
Answer: 24
Problem: <image>Question: There are $20$ cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all $20$ cities?

<image1>
Choices:
A. 65{,}000
B. 75{,}000
C. 85{,}000
D. 95{,}000
E. 105{,}000
Answer: D
Problem: <image>Question: Each of the points $A$, $B$, $C$, $D$, $E$, and $F$ in the figure below represent a different digit from 1 to 6. Each of the five lines shown passes through some of these points. The digits along the line each are added to produce 5 sums, one for each line. The total of the sums is $47$. What is the digit represented by $B$?

<image1>
Answer: 5
Problem: <image>Question: Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9$. What is the area of $ABCD?$

<image1>
Answer: 240
Problem: <image>Question: A game board consists of $64$ squares that alternate in color between black and white. The figure below shows square $P$ in the bottom and square $Q$ in the top row. A marker is placed at $P$. A step consists of moving the marker onto one of the adjoining white squares in the row above. How many $7$-step paths are there from $P$ to $Q$? (The figure shows a sample path.)

<image1>
Answer: 28
Problem: <image>Question: When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.
<image1>

For example, starting with an input of $N = 7$, the machine will output $3 \cdot 7 + 1 = 22$. Then if the output is repeatedly inserted into the machine five more times, the final output is $26$. $$ 7 \to 22 \to 11 \to 34  \to 17 \to 52 \to 26$$When the same 6-step process is applied to a different starting value of $N$, the final output is $1$. What is the sum of all such integers $N$? $$ N \to \_\_ \to \_\_ \to \_\_  \to \_\_ \to \_\_  \to 1$$
Choices:
A. 73
B. 74
C. 75
D. 82
E. 83
Answer: E
Problem: <image>Question: A large square region is paved with $n^2$ gray square tiles, each measuring $s$ inches on a side. A border $d$ inches wide surrounds each tile. The figure below shows the case for $n = 3$. When $n = 24$, the $576$ gray tiles cover $64\%$ of the area of the large square region. What is the ratio $\frac{d}{s}$ for this larger value of $n$?

<image1>
Choices:
A. $\frac{6}{25}$
B. $\frac{1}{4}$
C. $\frac{9}{25}$
D. $\frac{7}{16}$
E. $\frac{9}{16}$
Answer: A
Problem: <image>Question: Rectangles $R_1$ and $R_2,$ and squares $S_1,\,S_2,\,$ and $S_3,$ shown below, combine to form a rectangle that is $3322$ units wide and $2020$ units high. What is the side length of $S_2$ in units?
<image1>
Answer: 651
Problem: <image>Question: The Math Team designed a logo shaped like a multiplication symbol, shown below on a grid of 1-inch squares. What is the area of the logo in square inches?

<image1>
Answer: 10
Problem: <image>Question: The letter M in the figure below is first reflected over the line $q$ and then reflected over the line $p$. What is the resulting image?
<image1>

<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: One sunny day, Ling decided to take a hike in the mountains. She left her house at $8 \, \textsc{am}$, drove at a constant speed of $45$ miles per hour, and arrived at the hiking trail at $10 \, \textsc{am}$. After hiking for $3$ hours, Ling drove home at a constant speed of $60$ miles per hour. Which of the following graphs best illustrates the distance between Ling's car and her house over the course of her trip?

<image1>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: The arrows on the two spinners shown below are spun. Let the number $N$  equal 10 times the number on Spinner $A$, added to the number on Spinner $B$. What is the probability that $N$ is a perfect square number?
<image1>
Choices:
A. $\frac{1}{16}$
B. $\frac{1}{8}$
C. $\frac{1}{4}$
D. $\frac{3}{8}$
E. $\frac{1}{2} $
Answer: B
Problem: <image>Question: Laszlo went online to shop for black pepper and found thirty different black pepper options varying in weight and price, shown in the scatter plot below. In ounces, what is the weight of the pepper that offers the lowest price per ounce?

<image1>
Answer: 3
Problem: <image>Question: Mr. Ramos gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores.
<image1>
Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students $5$ extra points, which increased the median test score to $85$. What is the minimum number of students who received extra points?

(Note that the median test score equals the average of the $2$ scores in the middle if the $20$ test scores are arranged in increasing order.)
Answer: 4
Problem: <image>Question: The grid below is to be filled with integers in such a way that the sum of the numbers in each row and the sum of the numbers in each column are the same. Four numbers are missing. The number $x$ in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of $x$?
<image1>
Answer: 8
Problem: <image>Question: Steph scored $15$ baskets out of $20$ attempts in the first half of a game, and $10$ baskets out of $10$ attempts in the second half. Candace took $12$ attempts in the first half and $18$ attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?

<image1>
Answer: 9
Problem: <image>Question: A $\triangle$ or $\bigcirc$ is placed in each of the nine squares in a 3-by-3 grid. Shown below is a sample configuration with three $\triangle$s in a line.
<image1>

How many configurations will have three $\triangle$s in a line and three $\bigcirc$s in a line?
Answer: 84
Problem: <image>Question: The figure below shows a polygon $ABCDEFGH$, consisting of rectangles and right triangles. When cut out and folded on the dotted lines, the polygon forms a triangular prism. Suppose that $AH = EF = 8$ and $GH = 14$. What is the volume of the prism?

<image1>
Answer: 192
Problem: <image>Question: A square piece of paper is folded twice into four equal quarters, as shown below, then cut along the dashed line. When unfolded, the paper will match which of the following figures?
<image1>
<image2>
Choices:
A. A
B. B
C. C
D. D
E. E
Answer: E
Problem: <image>Question: The numbers from $1$ to $49$ are arranged in a spiral pattern on a square grid, beginning at the center. The first few numbers have been entered into the grid below. Consider the four numbers that will appear in the shaded squares, on the same diagonal as the number $7$. How many of these four numbers are prime?
<image1>
Answer: 3
Problem: <image>Question: The digits $2$, $0$, $2$, and $3$ are placed in the expression below, one digit per box. What is the maximum possible value of the expression?

<image1>
Answer: 9
Problem: <image>Question: A rectangle, with sides parallel to the $x-$axis and $y-$axis, has opposite vertices located at $(15, 3)$ and$(16, 5)$. A line is drawn through points $A(0, 0)$ and $B(3, 1)$. Another line is drawn through points $C(0, 10)$ and $D(2, 9)$. How many points on the rectangle lie on at least one of the two lines?
<image1>
Answer: 1
Problem: <image>Question: Malaika is skiing on a mountain. The graph below shows her elevation, in meters, above the base of the mountain as she skis along a trail. In total, how many seconds does she spend at an elevation between $4$ and $7$ meters?
<image1>
Answer: 8
Problem: <image>Question: The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded?
<image1>
Choices:
A. $\frac{1}{4}$
B. $\frac{11}{36}$
C. $\frac{1}{3}$
D. $\frac{19}{36}$
E. $\frac{5}{9}$
Answer: B
Problem: <image>Question: Along the route of a bicycle race, $7$ water stations are evenly spaced between the start and finish lines, as shown in the figure below. There are also $2$ repair stations evenly spaced between the start and finish lines. The $3$rd water station is located $2$ miles after the $1$st repair station. How long is the race in miles?
<image1>
Answer: 48
Problem: <image>Question: The letters $P$, $Q$, and $R$ are entered in a $20\times 20$ grid according to the pattern shown below. How many $P$s, $Q$s, and $R$s will appear in the completed table?

<image1>
Choices:
A. $132~\text{Ps}, 134~\text{Qs}, 134~\text{Rs}$
B. $133~\text{Ps}, 133~\text{Qs}, 134~\text{Rs}$
C. $133~\text{Ps}, 134~\text{Qs}, 133~\text{Rs}$
D. $134~\text{Ps}, 132~\text{Qs}, 134~\text{Rs}$
E. $134~\text{Ps}, 133~\text{Qs}, 133~\text{Rs}$
Answer: C
Problem: <image>Question: A regular octahedron has eight equilateral triangle faces with four faces meeting at each vertex. Jun will make the regular octahedron shown on the right by folding the piece of paper shown on the left. Which numbered face will end up to the right of $Q$?

<image1>
Answer: 1
Problem: <image>Question: An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is $\frac{2}{3}$ the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle?
<image1>
Choices:
A. 1:3
B. 3:8
C. 5:12
D. 7:16
E. 4:9
Answer: C
Problem: <image>Question: Each square in a $3 \times 3$ grid is randomly filled with one of the $4$ gray-and-white tiles shown below on the right.<image1>
What is the probability that the tiling will contain a large gray diamond in one of the smaller $2\times 2$ grids? Below is an example of one such tiling.
<image2>
Choices:
A. $\frac{1}{1024}$
B. $\frac{1}{256}$
C. $\frac{1}{64}$
D. $\frac{1}{16}$
E. $\frac{1}{4}$
Answer: C
Problem: <image>Question: Isosceles $\triangle$ $ABC$ has equal side lengths $AB$ and $BC$. In the figure below, segments are drawn parallel to $\overline{AC}$ so that the shaded portions of $\triangle$ $ABC$ have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height of $h$ of $\triangle$ $ABC$?

<image1>
Answer: 14.6
Problem: <image>Question: S-Corporation designs its logo by linking together $4$ semicircles along the diameter of a unit circle. Find the perimeter of the shaded portion of the logo.\n<image1>
Answer: $4 \pi$
Problem: <image>Question: Consider the figure <image1>, where every small triangle is equilateral with side length $1$. Compute the area of the polygon $ AEKS $.
Answer: $5 \sqrt{3}$
Problem: <image>Question: Let $\vartriangle ABC$ be an equilateral triangle with side length $M$ such that points $E_1$ and $E_2$ lie on side $AB$, $F_1$ and $F_2$ lie on side $BC$, and $G1$ and $G2$ lie on side $AC$, such that $$m = \overline{AE_1} = \overline{BE_2} = \overline{BF_1} = \overline{CF_2} = \overline{CG_1} = \overline{AG_2}$$and the area of polygon $E_1E_2F_1F_2G_1G_2$ equals the combined areas of $\vartriangle AE_1G_2$, $\vartriangle BF_1E_2$, and $\vartriangle CG_1F_2$. Find the ratio $\frac{m}{M}$.\n<image1>
Answer: $\frac{\sqrt{6}}{6}$
Problem: <image>Question: A group of aliens from Gliese $667$ Cc come to Earth to test the hypothesis that mathematics is indeed a universal language. To do this, they give you the following information about their mathematical system:\n\n$\bullet$ For the purposes of this experiment, the Gliesians have decided to write their equations in the same syntactic format as in Western math. For example, in Western math, the expression “$5+4$” is interpreted as running the “$+$” operation on numbers $5$ and $4$. Similarly, in Gliesian math, the expression $\alpha \gamma \beta$ is interpreted as running the “$\gamma $” operation on numbers $\alpha$ and $ \beta$.\n\n$\bullet$ You know that $\gamma $ and $\eta$ are the symbols for addition and multiplication (which works the same in Gliesian math as in Western math), but you don't know which is which. By some bizarre coincidence, the symbol for equality is the same in Gliesian math as it is in Western math; equality is denoted with an “$=$” symbol between the two equal values.\n\n$\bullet$ Two symbols that look exactly the same have the same meaning. Two symbols that are different have different meanings and, therefore, are not equal.\n\nThey then provide you with the following equations, written in Gliesian, which are known to be true:<image1>\n What is the human number equivalent of $๑$ ?
Answer: $\frac{1}{3}$
Problem: <image>Question: Right triangular prism $ABCDEF$ with triangular faces $\vartriangle ABC$ and $\vartriangle DEF$ and edges $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ has $\angle ABC = 90^o$ and $\angle EAB = \angle CAB = 60^o$ . Given that $AE = 2$, the volume of $ABCDEF$ can be written in the form $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.\n<image1>
Answer: 5
Problem: <image>Question: Alice is standing on the circumference of a large circular room of radius $10$. There is a circular pillar in the center of the room of radius $5$ that blocks Alice's view. The total area in the room Alice can see can be expressed in the form $\frac{m\pi}{n} +p\sqrt{q}$, where $m$ and $n$ are relatively prime positive integers and $p$ and $q$ are integers such that $q$ is square-free. Compute $m + n + p + q$. (Note that the pillar is not included in the total area of the room.)\n<image1>
Answer: 156
Problem: <image>Question: Let $A_1 = (0, 0)$, $B_1 = (1, 0)$, $C_1 = (1, 1)$, $D_1 = (0, 1)$. For all $i > 1$, we recursively define\n$$A_i =\frac{1}{2020} (A_{i-1} + 2019B_{i-1}),B_i =\frac{1}{2020} (B_{i-1} + 2019C_{i-1})$$$$C_i =\frac{1}{2020} (C_{i-1} + 2019D_{i-1}), D_i =\frac{1}{2020} (D_{i-1} + 2019A_{i-1})$$where all operations are done coordinate-wise.\n\nIf $[A_iB_iC_iD_i]$ denotes the area of $A_iB_iC_iD_i$, there are positive integers $a, b$, and $c$ such that $\sum_{i=1}^{\infty}[A_iB_iC_iD_i] = \frac{a^2b}{c}$, where $b$ is square-free and $c$ is as small as possible. Compute the value of $a + b + c$\n<image1>
Answer: 3031
Problem: <image>Question: Equilateral triangle $ABC$ has side length $2$. A semicircle is drawn with diameter $BC$ such that it lies outside the triangle, and minor arc $BC$ is drawn so that it is part of a circle centered at $A$. The area of the “lune” that is inside the semicircle but outside sector $ABC$ can be expressed in the form $\sqrt{p}-\frac{q\pi}{r}$, where $p, q$, and $ r$ are positive integers such that $q$ and $r$ are relatively prime. Compute $p + q + r$.\n<image1>
Answer: 10