1. Formulas
| Solid | Volume | Surface area |
|---|---|---|
| Cuboid | lbh | 2(lb + bh + hl); diagonal √(l² + b² + h²) |
| Cube | a³ | 6a²; diagonal a√3 |
| Cylinder | πr²h | CSA 2πrh; TSA 2πr(r + h) |
| Cone | ⅓πr²h | CSA πrl; TSA πr(r + l); l = √(r² + h²) |
| Sphere | (4/3)πr³ | 4πr² |
| Hemisphere | (2/3)πr³ | CSA 2πr²; TSA 3πr² |
| Frustum | ⅓πh(R² + r² + Rr) | CSA π(R + r)l |
| Hollow cylinder | πh(R² − r²) |
2. Useful facts
- 1 m³ = 1000 litres; 1 litre = 1000 cm³.
- Melting/recasting: volume stays the same. Number of small solids = big volume ÷ small volume.
- Cone, hemisphere and cylinder of same r and h = r: volumes 1 : 2 : 3.
- Largest cube in a sphere of diameter d: side d/√3.
3. Solved examples
(ASO 2015) Cone h 24, r 6 recast into a sphere → r³ = 216 → 6 cm.
(ASO 2012) Lead shots of diameter 3 cm from 9 × 11 × 12 cuboid → 1188 ÷ (99/7) = 84.
(ASO 2015) Longest pole in 10 × 10 × 5 room = √225 = 15 m.
Cylinder r 7, h 10: volume 1540 cm³, CSA 440 cm².
Tank 5 m × 4 m × 2 m holds 40 m³ = 40,000 L.
Solved examples by question type
Each type that comes in the ASO paper, worked step by step. Try it yourself first, then read the solution.
Basic level
Example 1. The total surface area of a cuboid 10 cm × 5 cm × 4 cm is:
(a) 220 cm² (b) 200 cm² (c) 120 cm² (d) 110 cm²
Answer: (a) 220 cm²
Solution: TSA = 2(lb + bh + hl) = 2(50 + 20 + 40) = 220 cm².
Example 2. A cube has a total surface area of 864 cm². Its volume is:
(a) 2,197 cm³ (b) 864 cm³ (c) 1,728 cm³ (d) 144 cm³
Answer: (c) 1,728 cm³
Solution: 6a² = 864 → a = 12. Volume = 12³ = 1728 cm³.
Example 3. The volume of a cylinder of radius 35 cm and height 7 cm is (π = 22/7):
(a) 27,720 cm³ (b) 26,950 cm³ (c) 3,850 cm³ (d) 1,540 cm³
Answer: (b) 26,950 cm³
Solution: πr²h = 22/7 × 35² × 7 = 26,950 cm³.
Example 4. The volume of a cuboid 6 cm × 11 cm × 21 cm is:
(a) 1,392 cm³ (b) 1,386 cm³ (c) 1,444 cm³ (d) 846 cm³
Answer: (b) 1,386 cm³
Solution: 6 × 11 × 21 = 1,386 cm³.
ASO exam level
Example 5. The curved surface area of a cone with slant height 25 cm and base radius 7 cm is (π = 22/7):
(a) 550 cm² (b) 154 cm² (c) 704 cm² (d) 1,100 cm²
Answer: (a) 550 cm²
Solution: CSA = πrl = 22/7 × 7 × 25 = 550 cm².
Example 6. The curved surface area of a cone with slant height 10 cm and base radius 7 cm is (π = 22/7):
(a) 220 cm² (b) 374 cm² (c) 440 cm² (d) 154 cm²
Answer: (a) 220 cm²
Solution: CSA = πrl = 22/7 × 7 × 10 = 220 cm².
Example 7. The height and slant height of a cone are 24 cm and 25 cm. Its volume is (π = 22/7):
(a) 3,696 cm³ (b) 1,232 cm³ (c) 616 cm³ (d) 15714.29 cm³
Answer: (b) 1,232 cm³
Solution: r² = 25² − 24² = 49. V = ⅓πr²h = ⅓ × 22/7 × 49 × 24 = 1,232 cm³.
Example 8. Find the curved surface area of a cylinder with radius 14 cm and height 20 cm (π = 22/7).
(a) 12,320 cm² (b) 1,760 cm² (c) 3,520 cm² (d) 1,804 cm²
Answer: (b) 1,760 cm²
Solution: V = πr²h = 12,320 cm³; CSA = 2πrh = 1,760 cm².
Example 9. The surface area of a sphere of radius 3.5 cm is (π = 22/7):
(a) 154 cm² (b) 115.5 cm² (c) 179.67 cm² (d) 77 cm²
Answer: (a) 154 cm²
Solution: SA = 4πr² = 4 × 22/7 × 12.25 = 154 cm².
Example 10. The height of a cone is twice the radius of its base. The ratio of the base area to the curved surface area is:
(a) 1 : 2 (b) 1 : √3 (c) √5 : 1 (d) 1 : √5
Answer: (d) 1 : √5
Solution: h = 2r → l = √(r² + 4r²) = √5 r. Base : CSA = πr² : πr·√5r = 1 : √5.
Example 11. The height of a cone is twice the radius of its base. The ratio of the base area to the curved surface area is:
(a) 1 : √5 (b) 1 : √3 (c) √5 : 1 (d) 1 : 2
Answer: (a) 1 : √5
Solution: h = 2r → l = √(r² + 4r²) = √5 r. Base : CSA = πr² : πr·√5r = 1 : √5.
Example 12. The shape of an ice-cream cone is a combination of:
(a) a sphere and a cylinder (b) a cylinder and a cone (c) two cones (d) a cone and a hemisphere
Answer: (d) a cone and a hemisphere
Solution: An ice-cream cone = a cone and a hemisphere.
Example 13. The shape of an ice-cream cone is a combination of:
(a) a cone and a hemisphere (b) a sphere and a cylinder (c) two cones (d) a cylinder and a cone
Answer: (a) a cone and a hemisphere
Solution: An ice-cream cone = a cone and a hemisphere.
Example 14. The surface area of a sphere of radius 21 cm is (π = 22/7):
(a) 5,544 cm² (b) 5,566 cm² (c) 2,772 cm² (d) 4,158 cm²
Answer: (a) 5,544 cm²
Solution: 4πr² = 4 × 22/7 × 21² = 5,544 cm².
Example 15. The total surface area of a solid hemisphere of radius 35 cm is (π = 22/7):
(a) 11,550 cm² (b) 7,700 cm² (c) 15,400 cm² (d) 11,572 cm²
Answer: (a) 11,550 cm²
Solution: 3πr² = 3 × 22/7 × 35² = 11,550 cm².
Example 16. The total surface area of a solid hemisphere of radius 28 cm is (π = 22/7):
(a) 4,928 cm² (b) 9,856 cm² (c) 7,392 cm² (d) 7,414 cm²
Answer: (c) 7,392 cm²
Solution: 3πr² = 3 × 22/7 × 28² = 7,392 cm².
Example 17. A tank is 8 m long, 5 m wide and 5 m deep. How many litres of water can it hold?
(a) 20,00,000 litres (b) 20,000 litres (c) 200 litres (d) 2,00,000 litres
Answer: (d) 2,00,000 litres
Solution: Volume = 200 m³; 1 m³ = 1000 litres → 2,00,000 litres.
Advanced level
Example 18. How many spherical lead shots of diameter 3 cm can be made from a cuboid of 11 cm × 12 cm × 9 cm?
(a) 90 (b) 84 (c) 168 (d) 42
Answer: (b) 84
Solution: Cuboid volume = 1188. One shot = 4/3 × 22/7 × (1.5)³ = 14.14. Number = 84.
Example 19. A metal sphere of radius 12 cm is melted and recast into 64 equal small spheres. The radius of each small sphere is:
(a) 4 cm (b) 0.19 cm (c) 6 cm (d) 3 cm
Answer: (d) 3 cm
Solution: Volume ∝ r³. Each small sphere = (1/64) of the volume → r = 12/∛64 = 3 cm.
Example 20. The diagonal of a cube is 10√3 cm. Its volume is:
(a) 3,000 cm³ (b) 1,010 cm³ (c) 1,000 cm³ (d) 600 cm³
Answer: (c) 1,000 cm³
Solution: Diagonal = a√3 → a = 10. Volume = 1,000 cm³.
Example 21. The diagonal of a cube is 11√3 cm. Its volume is:
(a) 1,331 cm³ (b) 1,342 cm³ (c) 726 cm³ (d) 3,993 cm³
Answer: (a) 1,331 cm³
Solution: Diagonal = a√3 → a = 11. Volume = 1,331 cm³.
Topic-wise practice for Mensuration: solids (Quant): Cube, cuboid, cylinder, cone, sphere, melting and recasting. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.
Question types covered
- Cube and cuboid: volume, surface area, diagonal
- Cylinder: volume, curved and total surface area
- Cone and frustum basics
- Sphere and hemisphere
- Melting and recasting
- Water level rise and filling tanks