Free practice questions/GRE General Test

GRE General Test — Geometry

30 free practice questions with full explanations.

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Question 1

What is the circumference of a circle with radius 9?

A) 9*pi
B) 18*pi
C) 27*pi
D) 81*pi
E) 36*pi

Show answer & explanation

Correct answer: B) 18*pi

Circumference = 2*pi*r = 2*pi*9 = 18*pi.

Question 2

Two similar rectangles have a ratio of corresponding sides of 3:5. If the smaller rectangle has an area of 27, what is the area of the larger rectangle?

A) 45
B) 60
C) 75
D) 81
E) 90

Show answer & explanation

Correct answer: C) 75

The ratio of areas equals the square of the ratio of sides: (5/3)^2 = 25/9. Area of larger rectangle = 27 x 25/9 = 75.

Question 3

A regular octagon has a perimeter of 64. What is the length of one side?

A) 6
B) 7
C) 8
D) 9
E) 10

Show answer & explanation

Correct answer: C) 8

A regular octagon has 8 equal sides. Side length = 64/8 = 8.

Question 4

A right triangle has legs of length 7 and 24. What is the length of the hypotenuse?

A) 24
B) 25
C) 26
D) 28
E) 31

Show answer & explanation

Correct answer: B) 25

By the Pythagorean theorem, hypotenuse = sqrt(7^2 + 24^2) = sqrt(49+576) = sqrt(625) = 25.

Question 5

What is the sum of the interior angles of a quadrilateral?

A) 540 degrees
B) 270 degrees
C) 720 degrees
D) 360 degrees
E) 180 degrees

Show answer & explanation

Correct answer: D) 360 degrees

Sum = (n-2)(180) = (4-2)(180) = 360 degrees.

Question 6

A circle has a radius of 5. What is its area (use pi = 3.14)?

A) 31.4
B) 15.7
C) 157
D) 25
E) 78.5

Show answer & explanation

Correct answer: E) 78.5

Area = pi r^2 = 3.14 x 25 = 78.5.

Question 7

A rectangle has a length of 18 and a width of 7. What is its area?

A) 25
B) 126
C) 116
D) 136
E) 50

Show answer & explanation

Correct answer: B) 126

Area = 18 x 7 = 126.

Question 8

A right triangle has legs of 6 and 8. What is the length of the hypotenuse?

A) 10
B) 14
C) 9
D) 12
E) 11

Show answer & explanation

Correct answer: A) 10

sqrt(6^2+8^2)=sqrt(36+64)=sqrt(100)=10.

Question 9

In a circle with center O, a central angle measures 60 degrees and the radius is 9. What is the length of the arc subtended by this angle?

A) 1.5*pi
B) 3*pi
C) 6*pi
D) 9*pi
E) 18*pi

Show answer & explanation

Correct answer: B) 3*pi

Arc length = (angle/360) x circumference = (60/360) x 2*pi*9 = (1/6) x 18*pi = 3*pi.

Question 10

A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?

A) 9
B) 10
C) 12
D) 14
E) 100

Show answer & explanation

Correct answer: B) 10

By the Pythagorean theorem, hypotenuse = sqrt(6^2 + 8^2) = sqrt(36+64) = sqrt(100) = 10.

Question 11

What is the area of a circle with radius 5?

A) 5*pi
B) 10*pi
C) 25*pi
D) 50*pi
E) 100*pi

Show answer & explanation

Correct answer: C) 25*pi

Area = pi*r^2 = pi*(5^2) = 25*pi.

Question 12

The interior angles of a triangle are in the ratio 2:3:4. What is the measure of the largest angle?

A) 40 degrees
B) 60 degrees
C) 80 degrees
D) 90 degrees
E) 100 degrees

Show answer & explanation

Correct answer: C) 80 degrees

The angles sum to 180 degrees. Total ratio parts = 2+3+4 = 9. Each part = 180/9 = 20 degrees. Largest angle = 4 x 20 = 80 degrees.

Question 13

A rectangular box has dimensions 4 x 5 x 6. What is its total surface area?

A) 74
B) 120
C) 128
D) 148
E) 240

Show answer & explanation

Correct answer: D) 148

Surface area = 2(lw + lh + wh) = 2(4x5 + 4x6 + 5x6) = 2(20 + 24 + 30) = 2(74) = 148.

Question 14

Two similar triangles have a ratio of corresponding sides of 2:3. If the area of the smaller triangle is 20, what is the area of the larger triangle?

A) 30
B) 40
C) 45
D) 60
E) 80

Show answer & explanation

Correct answer: C) 45

For similar figures, the ratio of areas equals the square of the ratio of corresponding sides: (3/2)^2 = 9/4. Area of larger triangle = 20 x 9/4 = 45.

Question 15

What is the area of a circle with diameter 14?

A) 14*pi
B) 28*pi
C) 49*pi
D) 98*pi
E) 196*pi

Show answer & explanation

Correct answer: C) 49*pi

Radius = diameter/2 = 7. Area = pi*r^2 = pi*49 = 49*pi.

Question 16

A cube has a volume of 125. What is the length of one edge of the cube?

A) 3
B) 4
C) 5
D) 25
E) 45

Show answer & explanation

Correct answer: C) 5

Volume of a cube = s^3 = 125. Since 5^3 = 125, s = 5.

Question 17

A right triangle has legs of length 9 and 12. What is the length of the hypotenuse?

A) 13
B) 14
C) 15
D) 16
E) 21

Show answer & explanation

Correct answer: C) 15

By the Pythagorean theorem, hypotenuse = sqrt(9^2 + 12^2) = sqrt(81+144) = sqrt(225) = 15.

Question 18

The angles of a quadrilateral measure x, x, 2x, and 2x degrees. What is the value of x?

A) 40
B) 50
C) 60
D) 70
E) 80

Show answer & explanation

Correct answer: C) 60

The interior angles of a quadrilateral sum to 360 degrees. x + x + 2x + 2x = 6x = 360. x = 60.

Question 19

The interior angles of a pentagon sum to how many degrees?

A) 360
B) 450
C) 540
D) 630
E) 720

Show answer & explanation

Correct answer: C) 540

The sum of interior angles of an n-sided polygon is (n-2) x 180. For a pentagon (n=5): (5-2) x 180 = 3 x 180 = 540 degrees.

Question 20

What is the area of a circle with circumference 20*pi?

A) 20*pi
B) 40*pi
C) 100*pi
D) 200*pi
E) 400*pi

Show answer & explanation

Correct answer: C) 100*pi

Circumference = 2*pi*r = 20*pi, so r = 10. Area = pi*r^2 = pi*100 = 100*pi.

Question 21

A right triangle has legs of length 8 and 15. What is the length of the hypotenuse?

A) 16
B) 17
C) 18
D) 20
E) 23

Show answer & explanation

Correct answer: B) 17

By the Pythagorean theorem, hypotenuse = sqrt(8^2 + 15^2) = sqrt(64+225) = sqrt(289) = 17.

Question 22

A cylinder has radius 4 and height 9. What is its volume, in terms of pi?

A) 36*pi
B) 72*pi
C) 108*pi
D) 144*pi
E) 288*pi

Show answer & explanation

Correct answer: D) 144*pi

Volume = pi*r^2*h = pi*16*9 = 144*pi.

Question 23

In a right triangle, one leg is 9 and the hypotenuse is 15. What is the length of the other leg?

A) 10
B) 11
C) 12
D) 13
E) 6

Show answer & explanation

Correct answer: C) 12

By the Pythagorean theorem: 9^2 + b^2 = 15^2 -> 81 + b^2 = 225 -> b^2 = 144 -> b = 12.

Question 24

A circle has an area of 49pi. What is its circumference?

A) 7pi
B) 14pi
C) 28pi
D) 49pi
E) 98pi

Show answer & explanation

Correct answer: B) 14pi

Area = pi*r^2 = 49pi -> r^2 = 49 -> r = 7. Circumference = 2*pi*r = 14pi.

Question 25

A rectangular box has dimensions 4, 5, and 6. What is its total surface area?

A) 120
B) 154
C) 140
D) 148
E) 160

Show answer & explanation

Correct answer: D) 148

Surface area = 2(lw + lh + wh) = 2(4x5 + 4x6 + 5x6) = 2(20+24+30) = 2(74) = 148.

Question 26

Two angles of a triangle measure 55 degrees and 65 degrees. What is the measure of the third angle?

A) 70 degrees
B) 50 degrees
C) 65 degrees
D) 55 degrees
E) 60 degrees

Show answer & explanation

Correct answer: E) 60 degrees

Triangle angles sum to 180. 180 - 55 - 65 = 60 degrees.

Question 27

A rectangle has a length of 22 and a width of 9. What is its area?

A) 176
B) 186
C) 208
D) 62
E) 198

Show answer & explanation

Correct answer: E) 198

Area = 22 x 9 = 198.

Question 28

A cylinder has a radius of 3 and a height of 10. What is its volume (use pi = 3.14)?

A) 94.2
B) 188.4
C) 282.6
D) 1130.4
E) 28.26

Show answer & explanation

Correct answer: C) 282.6

Volume = pi * r^2 * h = 3.14 * 3^2 * 10 = 3.14 * 9 * 10 = 282.6.

Question 29

A circle has a radius of 6. What is its area (use pi = 3.14)?

A) 37.68
B) 113.04
C) 18.84
D) 226.08
E) 452.16

Show answer & explanation

Correct answer: B) 113.04

Area = pi r^2 = 3.14 x 36 = 113.04.

Question 30

A right triangle has legs of 9 and 40. What is the length of the hypotenuse?

A) 39
B) 40
C) 41
D) 42
E) 49

Show answer & explanation

Correct answer: C) 41

sqrt(9^2+40^2)=sqrt(81+1600)=sqrt(1681)=41.

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