51 Volume & Surface Area aptitude questions and answers that are challenging and are designed to give you overall practice. In the answer, we have added a short description and formulas for solving the questions. Hope you will have a great experience and a lot of learning.
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Volume and Surface Area Formulas
Cube
Volume = a³
Total surface area = 6a²
Lateral surface area = 4a²
Diagonal = a√3
Cuboid
Volume = l × b × h
Total surface area = 2 (lb + bh + hl)
Lateral surface area = 2h (l + b)
Diagonal = √(l² + b² + h²)
Cylinder
Volume = πr²h
Curved surface area = 2πrh
Total surface area = 2πr (r + h)
Hollow cylinder or pipe
Volume of the material = πh (R² − r²)
Curved surface area = 2πh (R + r)
Cone
Volume = ⅓ πr²h
Curved surface area = πrl
Total surface area = πr (r + l)
Slant height l = √(r² + h²)
Sphere and hemisphere
Volume of a sphere = (4 ÷ 3) πr³
Surface area of a sphere = 4πr²
Volume of a hemisphere = (2 ÷ 3) πr³
Curved surface area of a hemisphere = 2πr²
Total surface area of a hemisphere = 3πr²
Prism and pyramid
Volume of a prism = Area of the base × Height
Lateral surface area of a prism = Perimeter of the base × Height
Volume of a pyramid = ⅓ × Area of the base × Height
Lateral surface area of a pyramid = ½ × Perimeter of the base × Slant height
Frustum of a cone
Volume = ⅓ πh (R² + r² + Rr)
Curved surface area = πl (R + r), where l = √(h² + (R − r)²)
Percentage change
If every dimension of a solid changes by x%, then
the surface area changes by 2x + (x² ÷ 100) percent
the volume changes by 3x + (3x² ÷ 100) + (x³ ÷ 10000) percent
If two solids are similar, the ratio of their surface areas is the square of the ratio of their sides, and the ratio of their volumes is the cube of it.
Units
1 m³ = 1000000 cm³
1 litre = 1000 cm³, so 1 m³ = 1000 litres
Take π as 22/7 when the radius is a multiple of 7, and as 3.14 otherwise.
Convert everything to one unit before you begin, never at the end.
51 Volume & Surface Area Aptitude Questions and Answers (Solved MCQs)
Volume & Surface Area Aptitude Test: Core Formulas (Cube, Cuboid, Cylinder, Cone, Sphere)
Question 1. The edge of a cube is 7 cm. Its total surface area is:
a) 196 cm²
b) 245 cm²
c) 294 cm²
d) 343 cm²
Answer:
c) 294 cm² — TSA of cube = 6a² = 6 × 7².
Question 2. A cuboid has length 10 cm, breadth 8 cm and height 6 cm. Its volume is:
a) 240 cm³
b) 360 cm³
c) 480 cm³
d) 600 cm³
Answer:
c) 480 cm³ — Volume = l × b × h = 10 × 8 × 6.
Question 3. A cylinder has radius 7 cm and height 10 cm. Its volume is: (Take π = 22/7)
a) 1440 cm³
b) 1540 cm³
c) 1640 cm³
d) 1760 cm³
Answer:
b) 1540 cm³ — V = πr²h = (22/7) × 7² × 10.
Question 4. A cylinder has radius 7 cm and height 10 cm. Its curved surface area is: (Take π = 22/7)
a) 308 cm²
b) 352 cm²
c) 440 cm²
d) 484 cm²
Answer:
c) 440 cm² — CSA = 2πrh = 2 × (22/7) × 7 × 10.
Question 5. The surface area of a sphere of radius 7 cm is: (Take π = 22/7)
a) 308 cm²
b) 616 cm²
c) 770 cm²
d) 924 cm²
Answer:
b) 616 cm² — SA = 4πr² = 4 × (22/7) × 7².
Question 6. The total surface area of a hemisphere of radius 7 cm is: (Take π = 22/7)
a) 308 cm²
b) 462 cm²
c) 616 cm²
d) 770 cm²
Answer:
b) 462 cm² — TSA(hemisphere) = 3πr².
Question 7. A cone has radius 7 cm and height 24 cm. Its curved surface area is: (Take π = 22/7)
a) 484 cm²
b) 550 cm²
c) 616 cm²
d) 704 cm²
Answer:
b) 550 cm² — l = √(7² + 24²) = 25; CSA = πrl = (22/7) × 7 × 25.
Question 8. A cone has radius 7 cm and height 24 cm. Its total surface area is: (Take π = 22/7)
a) 616 cm²
b) 650 cm²
c) 704 cm²
d) 770 cm²
Answer:
c) 704 cm² — TSA = πr(l + r) = (22/7) × 7 × (25 + 7).
Question 9. The volume of a cube is 512 cm³. The edge of the cube is:
a) 6 cm
b) 7 cm
c) 8 cm
d) 9 cm
Answer:
c) 8 cm — a³ = 512 ⇒ a = 8.
Question 10. A cuboid is 12 cm long, 9 cm broad, and 8 cm high. Its total surface area is:
a) 600 cm²
b) 624 cm²
c) 648 cm²
d) 672 cm²
Answer:
b) 624 cm² — TSA = 2(lb + bh + hl) = 2(108 + 72 + 96).
Question 11. The radius of a sphere is increased from 5 cm to 10 cm. Its surface area becomes how many times?
a) 2 times
b) 3 times
c) 4 times
d) 8 times
Answer:
c) 4 times — SA ∝ r²; (10/5)² = 4.
Question 12. The edge of a cube is doubled. Its volume becomes:
a) 2 times
b) 4 times
c) 6 times
d) 8 times
Answer:
d) 8 times — Volume ∝ a³; (2)³ = 8.
Question 13. A cylinder has diameter 14 cm and height 20 cm. Its curved surface area is: (Take π = 22/7)
a) 616 cm²
b) 704 cm²
c) 880 cm²
d) 968 cm²
Answer:
c) 880 cm²
Question 14. The volume of a cylinder is 3080 cm³. If radius = 7 cm, its height is: (Take π = 22/7)
a) 15 cm
b) 18 cm
c) 20 cm
d) 22 cm
Answer:
c) 20 cm — 3080 = (22/7) × 49 × h = 154 × h.
Question 15. A cone has radius 3 cm and slant height 5 cm. Its curved surface area is: (Take π = 3.14)
a) 37.68 cm²
b) 47.10 cm²
c) 54.30 cm²
d) 62.80 cm²
Answer:
b) 47.10 cm² — CSA = πrl = 3.14 × 3 × 5.
Question 16. The total surface area of a cube is 486 cm². Its edge is:
a) 7 cm
b) 8 cm
c) 9 cm
d) 10 cm
Answer:
c) 9 cm — 6a² = 486 ⇒ a² = 81.
Question 17. A hemisphere of radius 7 cm has volume: (Take π = 22/7)
a) 450 cm³
b) 539 cm³
c) 718.66 cm³
d) 1078 cm³
Answer:
c) 718.66 cm³.
Volume & Surface Area Aptitude Test: Composite Solids, Hollow Shapes & Costs
Question 18. A solid cuboid 10 cm × 8 cm × 6 cm is melted and recast into cubes of edge 2 cm. Number of cubes formed is:
a) 480
b) 240
c) 120
d) 60
Answer:
c) 120 — Total volume = 480; each cube volume = 8; 480 ÷ 8.
Question 19. A solid sphere of radius 6 cm is melted and recast into small spheres of radius 2 cm. Number of small spheres is:
a) 9
b) 18
c) 27
d) 36
Answer:
c) 27 — Number = (6³)/(2³) = 216/8.
Question 20. A cylinder of radius 7 cm and height 10 cm is melted to form a cone of radius 7 cm. Height of the cone is: (Take π = 22/7)
a) 20 cm
b) 25 cm
c) 30 cm
d) 35 cm
Answer:
c) 30 cm — πr²(10) = (1/3)πr²H ⇒ H = 30.
Question 21. A cone of radius 6 cm and height 8 cm is melted to form a sphere. The radius of the sphere is closest to:
a) 3.0 cm
b) 3.5 cm
c) 4.2 cm
d) 5.0 cm
Answer:
c) 4.2 cm — Volume conserved: (1/3)π×6²×8 = (4/3)πR³ ⇒ 96 = (4/3)R³ ⇒ R³ = 72 ⇒ R ≈ 4.16.
Question 22. A cylindrical pipe has inner radius 4 cm, outer radius 5 cm, and length 20 cm. Volume of metal used is: (Take π = 3.14)
a) 565.2 cm³
b) 565.5 cm³
c) 565.8 cm³
d) 566.0 cm³
Answer:
a) 565.2 cm³ — V = πh(R² − r²) = 3.14 × 20 × (25 − 16).
Question 23. A cube of edge 10 cm is painted on all faces and cut into cubes of edge 2 cm. Number of small cubes with exactly one face painted is:
a) 96
b) 72
c) 54
d) 48
Answer:
b) 72 — n = 10/2 = 5; one-face painted = 6(n−2)² = 6×3².
Question 24. A cube of edge 12 cm is painted on all faces and cut into cubes of edge 3 cm. Number of cubes with no face painted is:
a) 0
b) 1
c) 8
d) 16
Answer:
c) 8 — n = 12/3 = 4; none painted = (n−2)³ = 2³.
Question 25. A rectangular tank (open at top) is 8 m long, 6 m wide and 3 m deep. Area of sheet required to make it is:
a) 156 m²
b) 168 m²
c) 180 m²
d) 192 m²
Answer:
b) 168 m² — Open tank area = base + 4 walls = lw + 2lh + 2wh = 48 + 48 + 36.
Question 26. A cylindrical water tank has radius 3 m and height 7 m. Capacity in liters is: (Take π = 22/7)
a) 138,600 L
b) 176,000 L
c) 198,000 L
d) 207,900 L
Answer:
c) 198,000 L.
Question 27. The cost of painting the curved surface of a cylinder (r = 7 cm, h = 20 cm) at ₹0.50 per cm² is:
a) ₹220
b) ₹440
c) ₹484
d) ₹560
Answer:
b) ₹440 — CSA = 2πrh = 2 × (22/7) × 7 × 20 = 880; cost = 880 × 0.50.
Question 28. A solid cube of edge 6 cm is hollowed out to form a cubical box with thickness 1 cm on all sides. Volume of the hollow part is:
a) 64 cm³
b) 72 cm³
c) 80 cm³
d) 96 cm³
Answer:
a) 64 cm³ — Inner edge = 6 − 2×1 = 4; hollow volume = 4³.
Question 29. A hemispherical bowl of radius 7 cm is filled with water and poured into a cylindrical jar of radius 3.5 cm. Height of water in the jar is: (Take π = 22/7)
a) 7 cm
b) 9 cm
c) 10.67 cm
d) 14 cm
Answer:
c) 10.67 cm — (2/3)π×7³ = π×(3.5)²×h ⇒ h = (2/3)×343 / 12.25.
Question 30. A cone has base radius 6 cm and height 8 cm. It is filled with water and poured into a cylinder of radius 3 cm. Height of water in cylinder is:
a) 8 cm
b) 10 cm
c) 12 cm
d) 16 cm
Answer:
c) 12 cm — (1/3)π×6²×8 = π×3²×h ⇒ 96 = 9h.
Question 31. A cuboid of dimensions 15 cm × 10 cm × 6 cm is dipped in paint. Total painted area is:
a) 600 cm²
b) 660 cm²
c) 720 cm²
d) 780 cm²
Answer:
c) 720 cm² — TSA = 2(lb + bh + hl) = 2(150 + 60 + 90).
Question 32. A sphere and a cube have equal volumes. If the edge of the cube is 6 cm, radius of the sphere is closest to: (Take π = 3.14)
a) 3.2 cm
b) 3.7 cm
c) 4.1 cm
d) 4.8 cm
Answer:
b) 3.7 cm — 6³ = (4/3)πr³ ⇒ r³ = 216×3/(4π) ≈ 51.6 ⇒ r ≈ 3.7.
Question 33. A cylindrical log has radius 14 cm and length 1 m. Volume of the log is: (Take π = 22/7)
a) 30,800 cm³
b) 61,600 cm³
c) 92,400 cm³
d) 123,200 cm³
Answer:
b) 61,600 cm³ — h = 100 cm; V = πr²h = (22/7) × 14² × 100.
Volume & Surface Area Aptitude Test: Advanced Tricks, Ratios & Data Sufficiency
Question 34. The radius of a cylinder is tripled and height is doubled. The volume becomes:
a) 6 times
b) 9 times
c) 12 times
d) 18 times
Answer:
d) 18 times — V ∝ r²h ⇒ (3²)×2 = 18.
Question 35. The edge of a cube is increased by 20%. The total surface area increases by:
a) 20%
b) 40%
c) 44%
d) 72%
Answer:
c) 44% — TSA ∝ a² ⇒ (1.2)² = 1.44.
Question 36. The edge of a cube is decreased by 10%. The volume decreases by:
a) 10%
b) 19%
c) 27.1%
d) 30%
Answer:
c) 27.1% — Volume ∝ a³ ⇒ (0.9)³ = 0.729.
Question 37. A cube is cut into 125 equal smaller cubes. The ratio of total surface area of all small cubes to that of the original cube is:
a) 5:1
b) 10:1
c) 25:1
d) 125:1
Answer:
a) 5:1 — n = 5; TSA scales by n for equal partition: ratio = n = 5.
Question 38. A cuboid has volume 720 cm³. If l = 12 cm and b = 10 cm, then height is:
a) 4 cm
b) 5 cm
c) 6 cm
d) 8 cm
Answer:
c) 6 cm — h = 720 ÷ (12 × 10).
Question 39. A right circular cone has radius 6 cm and slant height 10 cm. Its height is:
a) 6 cm
b) 8 cm
c) 10 cm
d) 12 cm
Answer:
b) 8 cm — h = √(l² − r²) = √(100 − 36).
Question 40. A sphere has volume 288π cm³. Its radius is:
a) 3 cm
b) 4 cm
c) 6 cm
d) 8 cm
Answer:
c) 6 cm — (4/3)πr³ = 288π ⇒ r³ = 216.
Question 41. A hemispherical solid has volume 324π cm³. Its radius is:
a) 6 cm
b) 7 cm
c) 8 cm
d) 9 cm
Answer:
c) 8 cm — (2/3)πr³ = 324π ⇒ r³ = 486 ⇒ r ≈ 7.86 (closest 8).
Question 42. A cube of edge 9 cm is painted on all faces and cut into cubes of edge 1 cm. Number of cubes with exactly three faces painted is:
a) 6
b) 8
c) 12
d) 27
Answer:
b) 8 — Only the 8 corner cubes have 3 faces painted.
Question 43. A cube is painted and cut into n × n × n smaller cubes. Number of cubes with exactly two faces painted is:
a) 12(n − 2)
b) 6(n − 2)²
c) 8(n − 2)³
d) 12(n − 1)
Answer:
a) 12(n − 2) — Edge cubes excluding corners.
Question 44. A cylindrical tank has diameter 14 m and height 10 m. Its total inner surface area (including base) is: (Take π = 22/7)
a) 528 m²
b) 616 m²
c) 704 m²
d) 836 m²
Answer:
d) 836 m² — Inner area = CSA + base = 2πrh + πr² = 2×(22/7)×7×10 + (22/7)×49 = 440 + 154.
Question 45. DATA SUFFICIENCY: Find the volume of a cuboid.
I. Its length is 12 cm and breadth is 8 cm.
II. Its total surface area is 416 cm².
a) I alone sufficient
b) II alone sufficient
c) Both together sufficient, neither alone
d) Even together not sufficient
Answer:
c) Both together sufficient, neither alone
Question 46. DATA SUFFICIENCY: Find the radius of a sphere.
I. Its surface area is 616 cm².
II. Take π = 22/7.
a) I alone sufficient
b) II alone sufficient
c) Both together sufficient, neither alone
d) Even together not sufficient
Answer:
c) Both together sufficient, neither alone — Need π value to compute r from 4πr².
Question 47. A cube and a cuboid have equal total surface areas. Cube edge = 6 cm. Cuboid dimensions are 8 cm × 4 cm × h cm. The value of h is:
a) 4 cm
b) 5 cm
c) 6 cm
d) 7 cm
Answer:
b) 5 cm — TSA cube = 6×6²=216; TSA cuboid = 2(32 + 4h + 8h)=64+24h; set 64+24h=216.
Question 48. A cylinder and a cone have the same base radius 7 cm and the same height 12 cm. The ratio of their volumes is:
a) 1:2
b) 2:1
c) 3:1
d) 1:3
Answer:
c) 3:1 — Cylinder volume is 3 times cone volume for same r and h.
Question 49. A solid cylinder of radius 6 cm and height 9 cm is melted to form 8 identical cones, each of height 9 cm. The radius of each cone is:
a) 3 cm
b) 3.5 cm
c) (3√6)/2 cm
d) 4 cm
Answer:
c) (3√6)/2 cm — Volume conserved: π×6²×9 = 8×(1/3)π×r²×9 ⇒ 36 = 8r²/3 ⇒ r² = 13.5 = 27/2 ⇒ r = (3√6)/2.
Question 50. A sphere of radius 3 cm is placed inside a cylinder of radius 3 cm. Minimum height of the cylinder to fully contain the sphere is:
a) 3 cm
b) 4 cm
c) 5 cm
d) 6 cm
Answer:
d) 6 cm — Height must be the diameter = 2r.
Question 51. A cylinder has volume 616 cm³. If r = 7 cm, height is: (Take π = 22/7)
a) 3 cm
b) 4 cm
c) 5 cm
d) 6 cm
Answer:
b) 4 cm — V = (22/7)×49×h = 154h.
Aptitude Questions and Answers for other topics
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