🧭 Introduction
Geometry is a crucial part of the GRE Quantitative Reasoning section, testing your understanding of angles, shapes, and spatial logic. You’ll often face questions where a simple diagram hides multiple relationships — and speed plus accuracy make all the difference.
This post gives you 50 geometry practice problems divided by difficulty:
- 15 Easy
- 20 Medium
- 15 Hard
Each comes with clear, step-by-step solutions so you can not only memorize but understand the logic behind every problem.
📘 QUICK FORMULA RECAP
| Shape | Area Formula | Perimeter / Circumference | Key Concept |
|---|---|---|---|
| Triangle | ½ × base × height | a + b + c | 30–60–90 and 45–45–90 rules |
| Rectangle | l × w | 2(l + w) | Diagonals are equal |
| Circle | πr² | 2πr | 360° = 2π radians |
| Trapezium | ½ × (sum of parallel sides) × height | a + b + c + d | Bases parallel |
| Parallelogram | base × height | 2(a + b) | Opposite sides parallel |
| Cube | 6a² | 12a | All sides equal |
| Cylinder | 2πr(h + r) | — | Volume = πr²h |
🟢 SECTION A: EASY QUESTIONS (1–15)
Q1. Find the area of a triangle with base = 8 cm and height = 5 cm.
Solution:
Area = ½ × 8 × 5 = 20 cm²
Q2. A circle has a radius of 7 cm. Find its circumference.
Solution:
2πr = 2 × 3.14 × 7 = 43.96 cm
Q3. A rectangle has length 12 m and width 5 m. Find its diagonal.
Solution:
√(12² + 5²) = √169 = 13 m
Q4. A square’s area is 81 cm². Find its perimeter.
Solution:
√81 = 9 → 4 × 9 = 36 cm
Q5. A right triangle has hypotenuse = 10 and one side = 6. Find the other side.
Solution:
√(10² − 6²) = √64 = 8
Q6. Two complementary angles differ by 18°. Find both.
Solution:
x + (x + 18) = 90 → 2x = 72 → x = 36°, 54°
Q7. The diameter of a circle is 14 cm. Find its area.
Solution:
r = 7 → πr² = 3.14 × 49 = 153.86 cm²
Q8. The sides of a triangle are 3, 4, and 5. Find its area.
Solution:
Right triangle → ½ × 3 × 4 = 6 cm²
Q9. Find the perimeter of a regular hexagon with side 6 cm.
Solution:
6 × 6 = 36 cm
Q10. The diagonal of a square is 10√2. Find its side.
Solution:
Side = diagonal / √2 = 10 → 10 cm
Q11. The radius of a circle doubles. The area becomes?
Solution:
4 times larger.
Q12. A rectangle has perimeter 30 cm and length 9 cm. Find width.
Solution:
2(l + w) = 30 → 9 + w = 15 → w = 6 cm
Q13. Find the area of an equilateral triangle with side 4 cm.
Solution:
(√3 / 4) × a² = (1.732 / 4) × 16 = 6.93 cm²
Q14. Find the circumference of a semicircle with diameter 14 cm.
Solution:
(πr + 2r) = 3.14 × 7 + 14 = 35.98 cm
Q15. A square’s perimeter is 48 cm. Find its area.
Solution:
Side = 48 ÷ 4 = 12 → 12² = 144 cm²
🔵 SECTION B: MEDIUM QUESTIONS (16–35)
Q16. In a circle, central angle = 60°, radius = 6 cm. Find area of sector.
Solution:
(60 / 360) × πr² = (1/6) × 3.14 × 36 = 18.84 cm²
Q17. Find area of parallelogram with sides 8 cm, 6 cm, included angle 60°.
Solution:
ab sin(θ) = 8 × 6 × 0.866 = 41.6 cm²
Q18. Bases of a trapezium are 10 cm & 6 cm, height 4 cm. Find area.
Solution:
½ × (10 + 6) × 4 = 32 cm²
Q19. A triangle’s sides are 5, 12, and 13. Find area.
Solution:
Right triangle → ½ × 5 × 12 = 30 cm²
Q20. Find area of a rhombus with diagonals 10 cm & 8 cm.
Solution:
½ × 10 × 8 = 40 cm²
Q21. In ∆ABC, sides are 7, 8, 9. Find area (Heron’s formula).
Solution:
s = (7+8+9)/2 = 12 → √[12(12–7)(12–8)(12–9)] = √720 = 26.8 cm²
Q22. In coordinate geometry, find distance between (2, 3) and (−4, 6).
Solution:
√[(−4−2)² + (6−3)²] = √45 = 6.7 units
Q23. Find equation of a circle with center (3, −4), radius 5.
Solution:
(x−3)² + (y+4)² = 25
Q24. In a regular polygon, each interior angle = 120°. Find sides.
Solution:
(n−2)×180 / n = 120 → 180n − 360 = 120n → 60n = 360 → n = 6
Q25. A rectangular field 40 × 25 m has a diagonal path. Find its length.
Solution:
√(40² + 25²) = √2225 = 47.17 m
Q26. Find area of circle inscribed in square of side 14 cm.
Solution:
r = 7 → πr² = 3.14 × 49 = 153.86 cm²
Q27. Circumference of circle = 31.4 cm. Find radius.
Solution:
2πr = 31.4 → r = 5 cm
Q28. In ∆ABC, ∠A = 90°, AB = 9, AC = 12. Find BC.
Solution:
√(9² + 12²) = 15
Q29. Find coordinates of midpoint between (2, 5) and (10, −1).
Solution:
((2+10)/2, (5−1)/2) = (6, 2)
Q30. Equation of circle: x² + y² − 6x + 8y = 0. Find center & radius.
Solution:
(x−3)² + (y+4)² = 25 → Center (3, −4), r = 5
Q31. If radius of circle increases by 20%, find % increase in area.
Solution:
Area ∝ r² → 1.2² = 1.44 → 44% increase
Q32. Area of a triangle = 24, base = 8. Find height.
Solution:
24 = ½ × 8 × h → h = 6
Q33. Each side of equilateral triangle = 10 cm. Find height.
Solution:
(√3/2) × 10 = 8.66 cm
Q34. Diagonals of a rhombus are 16 cm and 12 cm. Find perimeter.
Solution:
Half-diagonals = 8 & 6 → side = √(8² + 6²) = 10 → 4 × 10 = 40 cm
Q35. A square and a circle have same perimeter. If side = 14 cm, find circle’s radius.
Solution:
4a = 2πr → 56 = 6.28r → r = 8.9 cm
🔴 SECTION C: HARD QUESTIONS (36–50)
Q36. A right circular cone has height 12 cm, slant height 13 cm. Find radius & total surface area.
Solution:
r² = 13² − 12² = 25 → r = 5
TSA = πr(l + r) = 3.14 × 5 × 18 = 282.6 cm²
Q37. A cylinder has radius 7 cm, height 10 cm. Find curved surface area.
Solution:
2πrh = 2 × 3.14 × 7 × 10 = 439.6 cm²
Q38. A cone and cylinder have same base and height. Ratio of their volumes?
Solution:
Cone : Cylinder = 1 : 3
Q39. Volume of sphere = 113.04 cm³. Find radius.
Solution:
(4/3)πr³ = 113.04 → r³ = 27 → r = 3 cm
Q40. In a triangle, sides are 13, 14, 15. Find area.
Solution:
s = 21 → √[21(21−13)(21−14)(21−15)] = √(21×8×7×6) = √7056 = 84 cm²
Q41. Equation of line joining (1, 2) and (3, 8)?
Solution:
Slope = (8−2)/(3−1) = 3 → y − 2 = 3(x − 1) → y = 3x − 1
Q42. Find coordinates where line y = 2x + 1 cuts x-axis.
Solution:
y = 0 → 0 = 2x + 1 → x = −0.5 → (−0.5, 0)
Q43. Height of cone = 9 cm, slant height = 15 cm. Find radius & volume.
Solution:
r² = 15² − 9² = 144 → r = 12
Volume = ⅓πr²h = ⅓ × 3.14 × 144 × 9 = 1356.5 cm³
Q44. Diameter of circle = 20 cm. Find length of longest chord.
Solution:
Longest chord = diameter = 20 cm
Q45. In coordinate plane, find slope of line joining (−2, 5) and (3, −1).
Solution:
(−1 − 5)/(3 + 2) = −6/5 = −1.2
Q46. The area of a semicircle = 77 cm². Find its diameter.
Solution:
(½)πr² = 77 → r² = 49 → r = 7 → Diameter = 14 cm
Q47. Find the area of a sector of circle with radius 10 cm and angle 108°.
Solution:
(108/360) × πr² = 0.3 × 3.14 × 100 = 94.2 cm²
Q48. The diagonal of a rectangle is 25 cm and one side is 24 cm. Find the other side.
Solution:
√(25² − 24²) = 7 → 7 cm
Q49. A sphere and cube have equal volumes. Cube side = 6 cm. Find sphere radius.
Solution:
a³ = (4/3)πr³ → 216 = 4.19r³ → r³ = 51.5 → r = 3.72 cm
Q50. A solid metal cube of side 5 cm is melted to form spheres of radius 2 cm. Find number of spheres.
Solution:
Volume cube = 125
Volume sphere = (4/3)πr³ = 33.5
125 ÷ 33.5 = 3.7 ≈ 3 spheres
🧠 GRE Geometry Quick Tips
- Draw diagrams — even for coordinate geometry; visual clarity speeds answers.
- Memorize ratios for special triangles.
- Check units — GRE loves to trick between cm² and m².
- Estimate answers before solving.
- Don’t panic on tough figures — break them into smaller parts.
✅ Conclusion
Mastering these 50 GRE Geometry Questions with detailed solutions prepares you to handle any diagram-based or reasoning-heavy geometry question on test day. Consistent practice will improve both your speed and accuracy — and that’s the key to scoring high on the quantitative section.