GRE Quantitative: GRE Quantitative: Geometry Practice Questions
Test yourself on GRE Quantitative: Geometry with 12 original GRE practice questions. Pick an answer to see instant feedback and a full explanation.
Free original practice questions for study purposes. Open Exam Prep is an independent study resource and is not affiliated with, endorsed by, or sponsored by the makers of GRE.
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Answer the questions below — you get instant feedback and a full explanation for each.
1. In a right triangle, the two legs have lengths 9 and 12. What is the length of the hypotenuse?
Explanation. By the Pythagorean theorem, hypotenuse = √(9² + 12²) = √(81 + 144) = √225 = 15. This is a 3-4-5 triangle scaled by 3.
2. The interior angles of a triangle are in the ratio 2:3:4. What is the measure of the largest angle?
Explanation. The parts sum to 2+3+4 = 9, and angles total 180°. Each part = 180/9 = 20°. The largest angle = 4 × 20 = 80°.
3. A circle has area 36π. What is its circumference?
Explanation. Area = πr² = 36π, so r² = 36 and r = 6. Circumference = 2πr = 2π(6) = 12π.
4. A rectangle has a perimeter of 40 and a length that is 3 times its width. What is its area?
Explanation. Let width = w, length = 3w. Perimeter = 2(w + 3w) = 8w = 40, so w = 5 and length = 15. Area = 5 × 15 = 75.
5. In the coordinate plane, what is the distance between points (−2, 3) and (4, −5)?
6. A cylinder has radius 3 and height 10. What is its volume?
Explanation. Volume of a cylinder = πr²h = π(3²)(10) = π(9)(10) = 90π.
7. Two parallel lines are cut by a transversal. One of the angles measures 110°. Which of the following CANNOT be the measure of another angle formed?
Explanation. When parallel lines are cut by a transversal, only two angle measures appear: the given angle (110°) and its supplement (70°). A 90° angle would only occur if the transversal were perpendicular, which is not the case here, so 90° cannot occur.
8. An equilateral triangle has a side length of 6. What is its area?
Explanation. Area of an equilateral triangle = (√3/4)s² = (√3/4)(36) = 9√3.
9. A square is inscribed in a circle of radius 5. What is the area of the square?
Explanation. The diagonal of the inscribed square equals the circle's diameter = 10. For a square, diagonal = s√2, so s√2 = 10 and s = 10/√2. Area = s² = 100/2 = 50.
10. In the xy-plane, line k has equation y = 2x − 3. Which line is perpendicular to line k?
Explanation. Perpendicular lines have slopes that are negative reciprocals. The slope of line k is 2, so a perpendicular line has slope −1/2. Only y = −(1/2)x + 7 has slope −1/2.
11. A right circular cone has a base radius of 4 and a height of 9. What is its volume?
Explanation. Volume of a cone = (1/3)πr²h = (1/3)π(16)(9) = (1/3)π(144) = 48π.
12. A circle has a chord that is 8 units long and lies 3 units from the center. What is the radius of the circle?
Explanation. The perpendicular from the center bisects the chord, creating a right triangle with legs 4 (half the chord) and 3 (distance to center), and the radius as hypotenuse. r = √(4² + 3²) = √25 = 5.
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FAQ
What geometry formulas must I memorize for the GRE?
Know the Pythagorean theorem and common triples (3-4-5, 5-12-13), special right triangles (30-60-90 and 45-45-90), area/perimeter of triangles, rectangles, and circles, volume/surface area of cylinders, cones, spheres, and boxes, the distance and slope formulas, and rules for parallel lines cut by a transversal.
Does the GRE provide a figure that is drawn to scale?
Not always. GRE figures are meant to provide information but are NOT necessarily drawn to scale unless stated. Coordinate-plane figures are drawn to scale. Rely on given measurements and stated relationships rather than appearance.
How should I approach geometry Quantitative Comparison questions?
Avoid assuming relationships from how a figure looks. Use given facts to set up equations, and test multiple valid configurations when a figure is underdetermined—if different valid shapes give different comparisons, the answer is 'cannot be determined.'