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Practice Geometry questions for the Graduate Record Examinations. Every question includes a full explanation of why the correct answer is right and why the tempting distractors are wrong.
15 questions available · medium difficulty · GRE · Free, no registration required
10 example questions with full explanations. Use the interactive practice above to work through the complete set.
Correct answer: A
Supplementary angles sum to 180°. Subtracting the given angle: 180° − 72° = 108°. OptionC(18°) results from confusing supplementary with complementary (which sum to 90°). Option A is the definition of a right angle, not the supplement of 72°. Option D is an arithmetic error.
Correct answer: B
The interior angles of any triangle sum to 180°. The third angle equals 180° − 54° − 83° = 43°. Option A (33°) likely results from subtracting from 170° instead of 180°. Option C (53°) may come from adding the two angles incorrectly as 127° instead of 137°. Option D (63°) is another arithmetic slip.
Correct answer: C
The area of a circle is πr², so with r = 7, the area is π(7²) = 49π. Option B (14π) equals the circumference (2πr), a common mix-up between area and circumference formulas. Option A (7π) omits both the squaring and the factor of 2. Option D (98π) may result from doubling the correct area.
Correct answer: D
The volume of a cube is s³, where s is the edge length. With s = 5, the volume is 5³ = 125 cubic centimeters. Option B (25) equals s², which is the area of one face, not the volume. Option A (15) equals 3s, a meaningless formula. Option C (150) equals the surface area of the cube (6s² = 6 × 25 = 150), a common confusion between volume and surface area.
Correct answer: A
The slope of line L is (3 − (−1))/(6 − 2) = 4/4 = 1. Since line M is perpendicular to L, its slope is the negative reciprocal: −1. Using point-slope form through (4, 1): y − 1 = −1(x − 4), which simplifies to y = −x + 5. Setting y = 0 gives x = 5. Choice A is the x-coordinate of the given point on M, a tempting but incorrect shortcut. Choices C and D result from sign errors when computing the perpendicular slope or the x-intercept.
Correct answer: B
The radius is the distance from center (−2, 3) to point (2, 6): r = √[(2−(−2))² + (6−3)²] = √[16 + 9] = √25 = 5. The standard form equation is (x + 2)² + (y − 3)² = 25. Choice A forgets to square the radius, writing r instead of r². Choice B reverses the signs of the center coordinates, a common error when reading the standard form. Choice D results from erroneously summing the coordinate differences as 4 + 3 = 7 instead of squaring and adding them.
Correct answer: C
Reflecting across the y-axis negates the x-coordinate while leaving y unchanged: (3, −5) → (−3, −5). Translating 4 units upward adds 4 to the y-coordinate: (−3, −5) → (−3, −1). Choice A applies the translation downward instead of upward. Choice B skips the reflection and only applies the translation. Choice D reflects across the x-axis instead of the y-axis and then translates, conflating the two types of reflection.
Correct answer: D
Since the circle is tangent to all four sides of a square with side 10, the diameter equals 10, so the radius is 5. The area of the square is 100 and the area of the inscribed circle is π(5²) = 25π. The shaded region is 100 − 25π. Choice C results from using the diameter (10) in place of π r², computing π(10) instead of π(5²). Choice B likely comes from using an incorrect radius of √5 or a partial squaring error. Choice A incorrectly uses the full side length 10 as the radius.
Correct answer: A
In a 45-45-90 triangle, the hypotenuse equals a leg multiplied by √2. With legs of length 7, the hypotenuse is 7√2. Choice A is just one leg. Choice C uses √3, which belongs to the 30-60-90 ratio, not the 45-45-90 ratio. Choice D would be twice the leg, which is not the correct relationship here.
Correct answer: B
In a 30-60-90 triangle, the side ratios are 1 : √3 : 2, corresponding to the sides opposite the 30°, 60°, and 90° angles respectively. The hypotenuse (opposite 90°) is always twice the shortest side (opposite 30°), so 2 × 5 = 10. Choice A misapplies the 45-45-90 ratio. Choice B confuses the longer leg with the hypotenuse. Choice D incorrectly triples the short side.