GRE Quiz: Polygons Circles

18 questions · exam conditions

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Polygons CirclesQuestion 1 of 18

A circle has radius $9$ . A sector of the circle has area $\frac{27\pi}{2}$ . What is the measure of the central angle of the sector, in degrees?

30^\circ

45^\circ

60^\circ

90^\circ

120^\circ

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GRE Quiz

GRE Quiz: Polygons Circles

Practice Polygons Circles in GRE with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Polygons Circles, giving you a quick way to practice the rules, question types, and explanations that matter most for GRE.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A circle has radius $9$ . A sector of the circle has area $\frac{27\pi}{2}$ . What is the measure of the central angle of the sector, in degrees?

$30^\circ$
$45^\circ$
$60^\circ$ (correct answer)
$90^\circ$
$120^\circ$

Explanation: This question tests understanding of circle sectors. The area of a sector is (θ/360) × πr², where θ is the central angle. Given r = 9 and area = 27π/2, set (θ/360) × 81π = 27π/2, so θ/360 = (27/2)/81 = 1/6, θ = 60°. This solves for θ using the given area. This justifies that the central angle is 60°, which is choice C. A tempting incorrect option is 90°, perhaps from using r instead of r². Another error could be dividing by π incorrectly, leading to 30° or 120°.

Question 2

A regular polygon has each interior angle measuring $165^\circ$ . How many sides does the polygon have?

12
18
20
24 (correct answer)
30

Explanation: This question tests knowledge of regular polygons. The interior angle is [(n-2) × 180°]/n. Setting this to 165° gives (n-2) × 180 = 165n, 180n - 360 = 165n, 15n = 360, n = 24. This solves the equation accurately. This justifies that the polygon has 24 sides, which is choice D. A tempting incorrect option is 18, perhaps from using exterior angle 15° incorrectly. Another common mistake is subtracting wrong, leading to 12 or 30.

Question 3

A sector of a circle has radius $6$ and arc length $4\pi$ . What is the measure of the central angle of the sector, in degrees?

$60^\circ$
$90^\circ$
$120^\circ$ (correct answer)
$150^\circ$
$240^\circ$

Explanation: This question tests understanding of circle sectors. The arc length is (θ/360) × 2πr. Given r = 6 and arc length 4π, (θ/360) × 12π = 4π, so θ/360 = 4/12 = 1/3, θ = 120°. This solves for θ using the arc formula. This justifies that the central angle is 120°, which is choice C. A tempting incorrect option is 90°, perhaps from using r instead of 2r in circumference. Another error could be using radians, leading to 60° or 240°.

Question 4

In a regular polygon, each interior angle is 5 times each exterior angle. How many diagonals does the polygon have?

12
35
54 (correct answer)
108

Explanation: Let each exterior angle be x; then the interior angle is 5x, and x + 5x = 180, so x = 30. Since an exterior angle of a regular polygon is 360 / n, you get n = 360 / 30 = 12. The number of diagonals is n(n - 3) / 2 = 12(9) / 2 = 54. Don't stop at 12; that's the number of sides, not diagonals.

Question 5

A sector with central angle 72° has area $5\pi$ . What is its arc length?

$2\pi$ (correct answer)
$5\pi$
$10\pi$
$25\pi$

Explanation: Since 72 degrees is one-fifth of a full circle, the sector area is one-fifth of the circle's area: (1/5)πr² = 5π, so r = 5. The arc length is also one-fifth of the circumference: (1/5)(2π*5) = 2π. The tempting wrong answer is 10π, the full circumference, because you must multiply by the sector fraction, not stop at r = 5.

Question 6

In a circle, AB is a diameter; C is on the circle. Arc AC not containing B is 100°. What is angle ABC?

$40^\circ$
$80^\circ$
$50^\circ$ (correct answer)
$100^\circ$

Explanation: Angle ABC is an inscribed angle because B is on the circle. Its intercepted arc is arc AC not containing B, which is 100 degrees, so the angle is half of that: 50 degrees. The tempting wrong answer is 100 degrees, but an inscribed angle is half the intercepted arc, not equal to it.

Question 7

Two externally tangent circles have centers 24 apart and radii in ratio 1:2. What is the larger circle's area?

$36\pi$
$64\pi$
$144\pi$
$256\pi$ (correct answer)

Explanation: Since the circles are externally tangent, the 24 is the sum of the two radii. With ratio 1:2, split 24 into thirds: radii are 8 and 16, so the larger radius is 16. Its area is 16 squared times pi = 256 pi. Don't use 64 pi; that is the area of the smaller circle, not the larger.

Question 8

A circle has circumference $18\pi$ . What is the radius of the circle?

18
9 (correct answer)
6
3
12

Explanation: This question tests the relationship between circumference and radius of a circle. The circumference formula is C = 2πr, where r is the radius. Given C = 18π, we solve for r: 18π = 2πr, which gives r = 18π/(2π) = 9. Therefore, the radius is 9 units. A common mistake is to confuse radius with diameter, which would incorrectly give 18 as the answer, or to divide by π instead of 2π.

Question 9

A circle has circumference $18\pi$ . What is the radius of the circle?

18
9 (correct answer)
6
3
$\frac{9}{2}$

Explanation: This question tests the relationship between circumference and radius of a circle. The circumference formula is C = 2πr, where r is the radius. Given C = 18π, we solve: 18π = 2πr, which gives r = 18π/(2π) = 9. Therefore, the radius is 9. Choice A (18) is a common error where students confuse radius with diameter, forgetting that diameter = 2×radius.

Question 10

A regular hexagon is inscribed in a circle of radius $10$ . What is the perimeter of the hexagon?

30
50
60 (correct answer)
100
120

Explanation: This question tests knowledge of polygons and circles. In a regular hexagon inscribed in a circle, each side equals the radius. Given radius 10, each side is 10, so the perimeter is 6 × 10 = 60. This applies the property of equilateral sides in a regular hexagon. This justifies that the perimeter is 60, which is choice C. A tempting incorrect option is 50, perhaps from miscounting the sides as 5. Another common mistake is using the circumference instead, leading to values like 120.

Question 11

A circle has radius $7$ centimeters. What is the circumference of the circle, in centimeters?

$7\pi$
$49\pi$
$14\pi$ (correct answer)
$28\pi$
$21\pi$

Explanation: This question tests the formula for the circumference of a circle. The circumference of a circle is given by C = 2πr, where r is the radius. With radius r = 7 centimeters, we calculate C = 2π(7) = 14π centimeters. Therefore, the circumference is 14π centimeters. A common mistake is confusing circumference with area, which would give πr² = 49π, leading to inCorrect answer B.

Question 12

A circle has diameter $10$ centimeters. What is the area of the circle, in square centimeters?

$100\pi$
$25\pi$ (correct answer)
$50\pi$
$10\pi$
$20\pi$

Explanation: This question tests the area formula for a circle when given the diameter. The area of a circle is A = πr², where r is the radius. Given diameter d = 10 centimeters, the radius r = d/2 = 5 centimeters. Therefore, A = π(5)² = 25π square centimeters. The area is 25π square centimeters. A common mistake is using the diameter directly in the formula instead of the radius, which would incorrectly give 100π.

Question 13

A convex polygon has $11$ sides. What is the sum of the interior angles of the polygon, in degrees?

1,440
1,800
1,260
1,620 (correct answer)
1,980

Explanation: This question tests the formula for the sum of interior angles in a polygon. The sum of interior angles of any convex polygon with n sides is (n-2) × 180°. For an 11-sided polygon, this equals (11-2) × 180° = 9 × 180° = 1,620°. Therefore, the sum of interior angles is 1,620°. A common error is to miscalculate the multiplication or to use n instead of (n-2) in the formula, which would give 1,980°.

Question 14

A regular octagon has perimeter $72$ . What is the length of each side?

8
12
9 (correct answer)
6
18

Explanation: This question tests the perimeter formula for regular polygons. A regular octagon has 8 equal sides, and perimeter equals the number of sides times the length of each side. Given perimeter P = 72, we find the side length s by dividing: s = P/8 = 72/8 = 9. Therefore, each side has length 9 units. A common error is to confuse the number of sides (an octagon has 8 sides, not 6 or 12), leading to incorrect answers like 12 or 6.

Question 15

A regular decagon has each interior angle measuring $x^\circ$ . What is the value of $x$ ?

144 (correct answer)
150
160
135
140

Explanation: This question tests knowledge of regular polygons. For a regular decagon (10 sides), each interior angle is [(10-2) × 180°]/10 = 1,440°/10 = 144°. This uses the standard formula for interior angles. This justifies that x = 144, which is choice A. A tempting incorrect option is 150, perhaps from estimating for a different n like 12. Another common mistake is calculating exterior as 36° and subtracting wrong, leading to 135 or 160.

Question 16

A regular octagon has all sides equal and all interior angles equal. What is the measure, in degrees, of each interior angle of the octagon?

90
135 (correct answer)
140
150
157.5

Explanation: This question tests knowledge of interior angles in regular polygons. The formula for each interior angle of a regular n-gon is (n-2)×180°/n. For an octagon, n = 8, so each interior angle = (8-2)×180°/8 = 6×180°/8 = 1080°/8 = 135°. Therefore, each interior angle of a regular octagon measures 135°. Choice A (90°) would be correct for a square, while choice D (150°) would apply to a regular 12-gon, making these common errors when students confuse polygon types.

Question 17

A regular pentagon is inscribed in a circle. The circle has radius $10$ . Which of the following equals the measure, in degrees, of the central angle subtending one side of the pentagon?

36
54
72 (correct answer)
108
144

Explanation: This question tests central angles in regular polygons inscribed in circles. When a regular n-gon is inscribed in a circle, the central angle subtending each side equals 360°/n. For a pentagon, n = 5, so each central angle = 360°/5 = 72°. Therefore, the central angle subtending one side is 72°. Choice D (108°) represents the interior angle of the pentagon itself, a common confusion between central angles and interior angles.

Question 18

In a circle with diameter $14$ , what is the length of an arc that subtends a central angle of $90^\circ$ ?

$7\pi$
$\frac{7\pi}{2}$ (correct answer)
$14\pi$
$\frac{49\pi}{2}$
$\frac{7\pi}{4}$

Explanation: This question tests the arc length formula for circles. Arc length is L = (θ/360°)×2πr, where θ is the central angle in degrees and r is the radius. With diameter 14, the radius is 7, and with θ = 90°, the arc length = (90°/360°)×2π×7 = (1/4)×14π = 7π/2. Therefore, the arc length is 7π/2. Choice C (14π) represents the full circumference, a common error when students forget to apply the angle fraction.