Surface Areas and Volumes: Class 10 Important Questions with Answers

20 MCQs (1 mark each)

1. The volume of a cube of edge 5 cm is
   (a) 25 cm³ (b) 125 cm³ (c) 150 cm³ (d) 75 cm³
2. The CSA of a hemisphere of radius 7 cm is
   (a) 154 cm² (b) 308 cm² (c) 462 cm² (d) 616 cm²
3. The slant height of a cone with r = 6, h = 8 is
   (a) 10 (b) 14 (c) 12 (d) 100
4. Two cubes of edge 4 cm are joined. TSA of the cuboid is
   (a) 96 cm² (b) 160 cm² (c) 192 cm² (d) 128 cm²
5. A cone and a cylinder have equal base and height. Volume ratio cone : cylinder
   (a) 1 : 2 (b) 1 : 3 (c) 3 : 1 (d) 2 : 3
6. The volume of a sphere of radius 3 cm (in terms of π) is
   (a) 36π (b) 12π (c) 27π (d) 9π
7. A capsule of total length 12 mm and diameter 4 mm has cylinder length
   (a) 8 mm (b) 10 mm (c) 4 mm (d) 6 mm
8. The TSA of a solid hemisphere of radius r is
   (a) 2πr² (b) 3πr² (c) 4πr² (d) πr²
9. A sphere and a hemisphere have the same radius. Ratio of their volumes
   (a) 1 : 2 (b) 2 : 1 (c) 4 : 3 (d) 3 : 2
10. If the radius of a sphere is doubled, its volume becomes
    (a) 2 times (b) 4 times (c) 8 times (d) 6 times
11. A toy is a cone of height 4 on a hemisphere of radius 3. Its total height
    (a) 7 (b) 10 (c) 4 (d) 5
12. Volume of a cylinder of r 7 cm, h 10 cm
    (a) 1540 cm³ (b) 440 cm³ (c) 154 cm³ (d) 4400 cm³
13. A cube of edge 7 cm has a hemisphere of diameter 7 cm on top. Extra area added
    (a) 38.5 cm² (b) 77 cm² (c) 154 cm² (d) 0
14. How many spheres of radius 1 cm can be made from a sphere of radius 4 cm?
    (a) 16 (b) 32 (c) 64 (d) 4
15. CSA of a cylinder is 88 cm² and h = 2 cm. Its radius
    (a) 14 cm (b) 7 cm (c) 3.5 cm (d) 2 cm
16. A cone of height 24 and radius 7 has CSA
    (a) 550 cm² (b) 528 cm² (c) 1056 cm² (d) 154 cm²
17. A hemispherical bowl of radius 21 cm holds
    (a) 9702 cm³ (b) 19404 cm³ (c) 4851 cm³ (d) 2425.5 cm³
18. The lateral surface area of a cube is 64 cm². Its edge
    (a) 4 cm (b) 8 cm (c) 16 cm (d) 2 cm
19. A cylinder and cone have radii 3 : 4, heights 2 : 3. Volume ratio
    (a) 9 : 8 (b) 8 : 9 (c) 3 : 4 (d) 1 : 3
20. A sphere is melted into a cylinder of the same radius. The cylinder's height is
    (a) r (b) 2r (c) $\frac{4}{3}r$ (d) $\frac{2}{3}r$

Answer key

1 (b) · 2 (b) · 3 (a) · 4 (b) · 5 (b) · 6 (a) · 7 (a) · 8 (b) · 9 (b) · 10 (c) · 11 (a) · 12 (a) · 13 (a) · 14 (c) · 15 (b) · 16 (a) · 17 (b) · 18 (a) · 19 (a) · 20 (c)

Hints: Q4 cuboid 8 × 4 × 4. Q11 4 + 3. Q13 gain πr² (bowl 2πr² − circle πr²) = 38.5. Q14 $\frac{64}{1}$. Q17 $\frac{2}{3}\cdot \frac{22}{7}\cdot 9261=19404$. Q20 $\frac{4}{3}\pi r^3=\pi r^{2}h$.

Assertion–Reason (10)

(a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.

1. A: The TSA of a toy (cone on hemisphere) is πrl + 2πr². R: The joined circular faces are not on the surface.
2. A: Scooping a hemisphere from a cube's face reduces its surface area. R: The volume of the solid reduces.
3. A: A cone and a cylinder with the same base and height have volumes in ratio 1 : 3. R: Volume of a cone is ⅓πr²h.
4. A: The slant height of a cone with r = 5, h = 12 is 13. R: l² = r² + h².
5. A: Two cubes of edge 10 cm joined have TSA 1200 cm². R: Each face is 100 cm².
6. A: Joining two hemispheres of equal radius along their bases gives a sphere. R: TSA of a sphere is 3πr².
7. A: Melting a solid and recasting keeps the volume the same. R: Mass and density do not change.
8. A: The volume of a hemisphere of radius 3 cm is 18π cm³. R: Volume of a hemisphere is ⅔πr³.
9. A: The CSA of a cylinder of r 7 cm and h 5 cm is 220 cm². R: CSA = 2πrh.
10. A: If the radius of a sphere is halved, its surface area becomes one fourth. R: Surface area is proportional to r².

Key: 1 (a) · 2 (d) · 3 (a) · 4 (a) · 5 (d) · 6 (c) · 7 (a) · 8 (a) · 9 (a) · 10 (a)

Q2: SA increases by πr² (bowl replaces a circle), so A is false. Q9: $2\cdot \frac{22}{7}\cdot 7\cdot 5=220$ ✓.

Very short answer (1 mark each)

1. Find the volume of a hemisphere of radius 21 cm. Ans. 19404 cm³
2. A cone has r 3 cm, h 4 cm. Find its slant height. Ans. 5 cm
3. Write the formula for TSA of a capsule. Ans. 2πrh + 4πr²
4. Find the TSA of a cube of edge 6 cm. Ans. 216 cm²
5. How many cubes of edge 2 cm fill a cube of edge 6 cm? Ans. 27
6. Volume of a cone with r 7, h 6 cm? Ans. 308 cm³
7. A 1 L bottle holds how many cm³? Ans. 1000
8. Find the CSA of a hemisphere of diameter 14 cm. Ans. 308 cm²

Short answer practice (3 marks each)

1. A toy is a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height is 15.5 cm. Find the volume of the toy.

2. A solid metallic sphere of radius 6 cm is melted and recast into a cone of radius 6 cm. Find the height of the cone.

3. Find the number of coins, 1.5 cm in diameter and 0.2 cm thick, to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm.

Long answer practice (5 marks each)

4. A solid is a cylinder with a hemisphere at one end and a cone at the other. The common radius is 3.5 cm, the cylinder is 10 cm and the cone 6 cm high. Find its total surface area and volume.

5. A bucket-shaped vessel is a cylinder of radius 7 cm and height 10 cm with a hemispherical bottom. Find the cost of painting its outer surface at ₹2 per 10 cm².

6. A tent is a cylinder of diameter 20 m and height 4.2 m surmounted by a cone of height 7.5 m (slant height 12.5 m). Find the canvas needed (no floor) and the air inside.