GRE Quantitative Reasoning Practice: Geometry
- Geometry
- 10 questions
- 15 minutes
These 10 problem-solving questions cover the geometry that appears in GRE Quantitative Reasoning: right and similar triangles, circles, polygons, boxes and lines in the coordinate plane. GRE figures are not necessarily drawn to scale, so work from the given numbers rather than from how a picture looks.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
A right triangle has legs of length 9 and 12.
What is the length of the altitude drawn from the right angle to the hypotenuse?
Correct answer: B
The hypotenuse is 15 and the area is 9 × 12 ÷ 2 = 54. Using the hypotenuse as the base, 15 × h ÷ 2 = 54, so h = 7.2. The value 7.5 is half the hypotenuse, which is the median to the hypotenuse, not the altitude.
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Question 2
A circle is inscribed in a square with side length 10, so the circle touches all four sides.
What is the area of the region inside the square but outside the circle?
Correct answer: C
The diameter equals the side, so the radius is 5 and the circle's area is 25π. Subtract from the square's area: 100 − 25π. Using 10 as the radius gives 100π, which is larger than the square itself.
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Question 3
What is the measure of each interior angle of a regular octagon?
Correct answer: E
The interior angles of any octagon add up to (8 − 2) × 180° = 1,080°, and 1,080° ÷ 8 = 135°. The value 45° is the exterior angle (360° ÷ 8), not the interior angle.
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Question 4
In the xy-plane, point P is at (−2, 3) and point Q is at (4, −5).
What is the distance between P and Q?
Correct answer: A
The horizontal change is 6 and the vertical change is 8, so the distance is the square root of 36 + 64 = 100, which is 10. Adding the changes (6 + 8 = 14) measures a path along the grid, not the straight-line distance.
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Question 5
A circle has radius 9. An arc of the circle has a central angle of 80°.
What is the length of the arc?
Correct answer: D
The arc is 80/360 = 2/9 of the circumference 18π, and 2/9 × 18π = 4π. The value 18π is the whole circumference.
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Question 6
Two sides of a triangle have lengths 7 and 11. The third side has a length that is an integer.
How many different lengths are possible for the third side?
Correct answer: B
The third side must be greater than 11 − 7 = 4 and less than 11 + 7 = 18, so it can be any integer from 5 to 17: that is 17 − 5 + 1 = 13 values. Subtracting 17 − 5 without adding 1 gives 12, and allowing 4 and 18 would give 15, though both make a flat triangle.
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Question 7
A closed rectangular box measures 3 inches by 4 inches by 12 inches.
What is the total surface area of the box, in square inches?
Correct answer: E
The three different faces have areas 12, 36 and 48, and each appears twice: 2 × (12 + 36 + 48) = 192. The value 96 forgets that every face has an opposite twin, and 144 is the volume.
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Question 8
Line k passes through the point (1, 2) and is perpendicular to the line y = 3x − 4.
What is the y-intercept of line k?
Correct answer: A
A perpendicular line has slope −13. From y − 2 = −13(x − 1), setting x = 0 gives y = 2 + 13 = 73. Using slope +13 (reciprocal without the sign change) gives 53.
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Question 9
An isosceles triangle has two sides of length 10 and a base of length 12.
What is the area of the triangle?
Correct answer: C
The altitude to the base splits it into halves of 6, so the height is the square root of 10² − 6² = 64, which is 8. The area is 12 × 8 ÷ 2 = 48. The value 60 wrongly uses the slanted side 10 as the height.
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Question 10
Two triangles are similar, and the ratio of their corresponding sides is 2 : 3. The smaller triangle has an area of 24.
What is the area of the larger triangle?
Correct answer: D
Areas of similar figures scale with the square of the side ratio, so the larger area is 24 × (3/2)² = 24 × 9/4 = 54. Multiplying by 3/2 only gives 36, which treats area like length.
Questions you leave blank are not counted as wrong.
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