Keerthana Classes

Maths ch12 surface areas and volumes ·

October 04, 2026

37 marks. Read the notes, then tap Start test. Answers and your score appear only after you submit.

Quick notes

Cylinder: CSA = 2πrh, TSA = 2πr(r + h), Volume = πr²h.
Cone: l = √(r² + h²), CSA = πrl, Volume = (1/3)πr²h.
Sphere: SA = 4πr², Volume = (4/3)πr³.
Hemisphere: CSA = 2πr², TSA = 3πr², Volume = (2/3)πr³.
Cube: TSA = 6a², Volume = a³. Cuboid: TSA = 2(lb + bh + hl), Volume = lbh.

Combination of solids: the surface area includes only the parts you can see from outside (joined faces are left out). The volume is the sum of the volumes of the parts.

Part A: Multiple choice 16 questions

1. The curved surface area of a cylinder is

  1. πr²h
  2. 2πrh
  3. 2πr(r + h)
  4. πrl

Unroll it into a rectangle 2πr by h.

2. The volume of a cone is

  1. (1/3)πr²h
  2. πr²h
  3. (2/3)πr³
  4. πrl

One third of the cylinder with the same base and height.

3. The total surface area of a solid hemisphere is

  1. 2πr²
  2. 4πr²
  3. 3πr²
  4. πr²

Curved part 2πr² plus flat circle πr².

4. The volume of a sphere is

  1. 4πr²
  2. (2/3)πr³
  3. πr³
  4. (4/3)πr³

Standard formula.

5. Two cubes of volume 64 cm³ each are joined end to end. The surface area of the cuboid formed is

  1. 160 cm²
  2. 192 cm²
  3. 128 cm²
  4. 96 cm²

Cuboid 8 × 4 × 4: 2(32 + 16 + 32) = 160 cm².

6. A toy is a cone on a hemisphere. Its total surface area is

  1. TSA cone + TSA hemisphere
  2. CSA cone + CSA hemisphere
  3. CSA cone + TSA hemisphere
  4. volume of both

The joined circle is hidden, so only curved surfaces count.

7. The volume of a solid made of a cylinder and a hemisphere is

  1. the larger volume
  2. the difference of volumes
  3. the sum of the volumes
  4. the product of volumes

Volumes simply add up.

8. The slant height of a cone with r = 3 cm and h = 4 cm is

  1. 7 cm
  2. 12 cm
  3. 1 cm
  4. 5 cm

√(9 + 16) = 5 cm.

9. The volume of a hemisphere of radius 3 cm is

  1. 18π cm³
  2. 36π cm³
  3. 9π cm³
  4. 27π cm³

(2/3)π × 27 = 18π cm³.

10. The curved surface area of a cone with r = 7 cm and l = 10 cm (π = 22/7) is

  1. 154 cm²
  2. 220 cm²
  3. 440 cm²
  4. 70 cm²

(22/7) × 7 × 10 = 220 cm².

11. A capsule is 14 mm long with diameter 5 mm (a cylinder with two hemispherical ends). The cylindrical part is

  1. 14 mm
  2. 5 mm
  3. 9 mm
  4. 4 mm

14 − 2 × 2.5 = 9 mm.

12. The volume of a cylinder with r = 7 cm and h = 10 cm (π = 22/7) is

  1. 440 cm³
  2. 770 cm³
  3. 154 cm³
  4. 1540 cm³

(22/7) × 49 × 10 = 1540 cm³.

13. The surface area of a sphere of radius 7 cm (π = 22/7) is

  1. 616 cm²
  2. 154 cm²
  3. 308 cm²
  4. 1437 cm²

4 × (22/7) × 49 = 616 cm².

14. A 7 cm cube has a hemisphere of diameter 7 cm on top. Its surface area is

  1. 294 cm²
  2. 332.5 cm²
  3. 371 cm²
  4. 255.5 cm²

6 × 49 − πr² + 2πr² = 294 + 38.5 = 332.5 cm².

15. The volume of a cone with r = 3 cm and h = 7 cm (π = 22/7) is

  1. 198 cm³
  2. 132 cm³
  3. 66 cm³
  4. 22 cm³

(1/3) × (22/7) × 9 × 7 = 66 cm³.

16. When a cone is placed on a cylinder with the same base, the joined circular area is

  1. counted twice
  2. added to the volume
  3. counted once
  4. not included in the surface area

Hidden faces are not part of the outer surface.

Part B: Fill in the blanks 8 blanks

17. Volume of a cylinder = πr²[[h]].
18. Curved surface area of a hemisphere = [[2]]πr².
19. For a cone with r = 6 cm and h = 8 cm, the slant height is [[10]] cm.
20. Volume of a hemisphere = [[2/3]]πr³.
21. The volume of a cube of side 5 cm is [[125]] cm³.
22. The surface area of a sphere of radius 3.5 cm (π = 22/7) is [[154]] cm².
23. The volume of a cone is [[one-third|1/3|one third]] of the volume of a cylinder with the same base and height.
24. The total surface area of a cube of side 4 cm is [[96]] cm².

Part C: Questions and answers 9 questions

25. Two cubes, each of volume 64 cm³, are joined end to end. Find the surface area of the resulting cuboid.

Side = 4 cm, so the cuboid is 8 × 4 × 4.
SA = 2(8×4 + 4×4 + 4×8) = 2(32 + 16 + 32) = 160 cm².

26. A vessel is a hollow hemisphere topped by a hollow cylinder. The diameter is 14 cm and the total height is 13 cm. Find the inner surface area.

r = 7, cylinder height = 13 − 7 = 6.
Area = 2πrh + 2πr² = 264 + 308 = 572 cm².

27. A toy is a cone of radius 3.5 cm on a hemisphere of the same radius. The total height is 15.5 cm. Find its total surface area.

Cone height = 12, l = √(144 + 12.25) = 12.5.
SA = πrl + 2πr² = 137.5 + 77 = 214.5 cm².

28. A cubical block of side 7 cm is surmounted by a hemisphere of the greatest possible diameter. Find the surface area of the solid.

Hemisphere r = 3.5. SA = 6 × 49 − πr² + 2πr² = 294 + 38.5 = 332.5 cm².

29. A solid is a cone on a hemisphere, both of radius 1 cm, and the cone's height equals its radius. Find the volume in terms of π.

Cone = (1/3)π(1)(1) = π/3. Hemisphere = (2/3)π.
Total = π cm³.

30. A capsule is a cylinder with two hemispherical ends. Its total length is 14 mm and diameter 5 mm. Find its surface area.

r = 2.5, cylinder h = 9.
SA = 2πrh + 4πr² = 990/7 + 550/7 = 220 mm².

31. A pen stand is a 15 cm × 10 cm × 3.5 cm cuboid with four conical holes of radius 0.5 cm and depth 1.4 cm. Find the volume of wood.

Cuboid = 525 cm³. Each cone = (1/3)(22/7)(0.25)(1.4) ≈ 0.367 cm³; four cones ≈ 1.47 cm³.
Wood ≈ 523.53 cm³.

32. A tent is a cylinder of height 2.1 m and diameter 4 m with a conical top of slant height 2.8 m. Find the canvas needed.

r = 2. Canvas = 2πrh + πrl = 26.4 + 17.6 = 44 m².

33. An iron pole is a cylinder of height 220 cm and base diameter 24 cm, with a cylinder of height 60 cm and radius 8 cm on top. Find its mass if 1 cm³ of iron is 8 g (π = 3.14).

Volume = π(144 × 220 + 64 × 60) = 35520π = 111532.8 cm³.
Mass = 892262.4 g ≈ 892.26 kg.

Part D: Word puzzles 4 puzzles

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