MCQs for Surface Areas and Volumes: Class 10 Maths

Surface Areas and Volumes is an important chapter in Class 10 Maths that focuses on calculating the area and volume of three-dimensional shapes such as cylinders, cones, spheres, cubes, and cuboids. Practicing MCQs for Surface Areas and Volumes helps students strengthen conceptual understanding, improve calculation speed, and prepare effectively for school and board examinations.

Table of Contents

• MCQS and Answers for Surface Areas and Volumes for Class 10
• Practice Questions for Surface Areas and Volumes for Class 10

MCQS and Answers for Surface Areas and Volumes for Class 10

1. If the radius of a sphere is doubled, the ratio of the volume of the new sphere to the old sphere is:

(A) 2 : 1 (B) 4 : 1

(C) 8 : 1 (D) 6 : 1

Answer: (C) Volume ∝ r³. If radius doubles, volume becomes 2³ = 8 times. Ratio = 8 : 1.

 

2. The total surface area of a solid hemisphere of radius r is:

(A) 2πr² (B) 3πr² 

(C) 4πr² (D) πr²

Answer: (B)

TSA = CSA of hemisphere + area of flat base = 2πr² + πr² = 3πr².

 

3. A metallic sphere of radius 10.5 cm is melted and then recast into small cones, each of radius 3.5 cm and height 3 cm. The number of cones formed is:

(A) 107 (B) 112

(C) 126 (D) 135

Answer: (C) 

Volume of sphere = (4/3)π(10.5)³. 

Volume of cone = (1/3)π(3.5)²(3). 

⇒ Number = sphere vol / cone vol = 126.

 

4. The slant height of a frustum of a cone is 4 cm. If the perimeters of its circular ends are 18 cm and 6 cm, the curved surface area of the frustum is:

(A) 48 cm² (B) 48 cm² 

(C) 36 cm² (D) 56 cm²

Answer: (B)

CSA = π(R+r)l. 2πR = 18 

⇒ R = 9/π. 2πr = 6 

⇒ r = 3/π. 

CSA = π × (12/π) × 4 = 48 cm².

 

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

(A) 128 cm² (B) 256 cm²

(C) 160 cm² (D) 192 cm²

Answer: (C)

Each cube has edge 4 cm. 

Resulting cuboid: 8 × 4 × 4 cm. 

⇒ TSA = 2(8×4 + 4×4 + 8×4) = 2(32+16+32) = 2×80 = 160 cm².

 

6. A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is:

(A) 1 : 2 : 3 (B) 2 : 1 : 3

(C) 3 : 2 : 1 (D) 1 : 3 : 2

Answer: (A) 

With equal base radius r and height r: cone = (1/3)πr³, hemisphere = (2/3)πr³, cylinder = πr³. 

⇒ Ratio = 1 : 2 : 3.

 

7. The total surface area of a hemisphere of radius r is:

(a) 2πr² (b) 3πr²

(c) 4πr² (d) πr²

Answer: (b) 3πr²

A hemisphere has two surfaces: the curved dome and the flat circular base. CSA = 2πr² and base area = πr². So TSA = 2πr² + πr² = 3πr².

 

8. The slant height of a cone with height 12 cm and base radius 5 cm is:

(a) 11 cm (b) 13 cm

(c) 15 cm (d) 17 cm

Answer: (b) 13 cm

Using the Pythagorean theorem: l = √(h² + r²) = √(144 + 25) = √169 = 13 cm.

 

9. If two solid hemispheres of the same base radius r are joined together along their bases, the curved surface area of the new solid is:

(a) 3πr² (b) 4πr²

(c) 6πr² (d) 2πr²

Answer: (b) 4πr²

When two hemispheres are joined at their flat bases, they form a complete sphere. The CSA of a sphere = 4πr². The flat bases are now internal and not part of the outer surface.

 

10. A toy is in the shape of a cone mounted on a hemisphere. The radius of the base of the cone and hemisphere is 7 cm. The height of the cone is 24 cm. The slant height of the cone is:

(a) 20 cm (b) 25 cm

(c) 27 cm (d) 30 cm

Answer: (b) 25 cm

l = √(h² + r²) = √(576 + 49) = √625 = 25 cm.

 

11. A toy is in the shape of a cone mounted on a hemisphere. The radius of the base of the cone and hemisphere is 7 cm. The height of the cone is 24 cm.the total surface area of the toy is: (π = 22/7)

(a) 770 cm² (b) 858 cm²

(c) 924 cm² (d) 1034 cm²

Answer: (b) 858 cm²

TSA of toy = CSA of cone + CSA of hemisphere = πrl + 2πr²

= (22/7) × 7 × 25 + 2 × (22/7) × 49

= 550 + 308 = 858 cm²

 

12. Two cubes, each of volume 64 cm³, are joined end to end. The surface area of the resulting cuboid is:

(a) 128 cm² (b) 96 cm²

(c) 160 cm² (d) 192 cm²

Answer: (c) 160 cm²

Edge of each cube = ∛64 = 4 cm. The new cuboid is 8 cm × 4 cm × 4 cm.

SA = 2(lb + bh + lh) = 2(32 + 16 + 32) = 2 × 80 = 160 cm²

 

13. A cylinder and a cone have the same base radius and same height. The ratio of their volumes is:

(a) 1 : 3 (b) 3 : 1

(c) 1 : 2 (d) 2 : 1

Answer: (b) 3 : 1

Volume of cylinder = πr²h; Volume of cone = (1/3)πr²h. Ratio = πr²h : (1/3)πr²h = 3 : 1.

 

14. A cylinder, a cone and a hemisphere have equal base and the same height. The ratio of their volumes is:

(a) 1 : 2 : 3 (b) 3 : 1 : 2

(c) 3 : 2 : 1 (d) 1 : 3 : 2

Answer: (b) 3 : 1 : 2

Let radius = r, height = r (hemisphere's height = radius).

V_cylinder = πr³; V_cone = πr³/3; V_hemisphere = 2πr³/3

Ratio = 1 : 1/3 : 2/3 = 3 : 1 : 2

 

15. A metallic sphere of radius 6 cm is melted and recast into small spheres, each of radius 2 cm. The number of small spheres formed is:

(a) 9 (b) 18

(c) 27 (d) 36

Answer: (c) 27

Volume is conserved during melting.

n × (4/3)π(2)³ = (4/3)π(6)³

n = 6³ / 2³ = 216/8 = 27

 

16. A solid sphere of radius r is melted and recast into a solid cone of the same radius r. The height of the cone will be:

(a) r (b) 2r

(c) 3r (d) 4r

Answer: (d) 4r

Volume of sphere = (4/3)πr³; Volume of cone = (1/3)πr²h

Setting equal: (1/3)πr²h = (4/3)πr³

h = 4r

 

17. A solid metallic cylinder of base radius 3 cm and height 5 cm is melted to form cones, each of height 1 cm and base radius 1 mm. The number of cones is:

(a) 450 (b) 1350

(c) 4500 (d) 13500

Answer: (d) 13500

V_cylinder = π × 9 × 5 = 45π cm³

V_one_cone = (1/3)π × (0.1)² × 1 = π/300 cm³ (radius = 1 mm = 0.1 cm)

n = 45π ÷ (π/300) = 45 × 300 = 13500

 

18. If the diameter of a sphere is decreased by 25%, by what percentage does the curved surface area decrease?

(a) 25% (b) 37.5%

(c) 43.75% (d) 50%

Answer: (c) 43.75%

New radius = 0.75r. New CSA = 4π(0.75r)² = 4π × 0.5625r²

Decrease % = (1 − 0.5625) × 100 = 43.75%

 

19. The slant height of the frustum of a cone having radii of two ends as 5 cm and 2 cm, and height 4 cm, is:

(a) √26 cm (b) 5 cm

(c) √65 cm (d) 25 cm

Answer: (b) 5 cm

l = √[h² + (R − r)²] = √[16 + 9] = √25 = 5 cm

 

20. The curved surface area of a frustum with slant height l and radii R and r is:

(a) π(R + r)l (b) π(R − r)l

(c) 2π(R + r)l (d) π(R² + r²)

Answer: (a) π(R + r)l

 

21. A bucket is in the form of a frustum with top radius 18 cm, bottom radius 12 cm, and height 12 cm. Its volume is: (π = 3.14)

(a) 5765.7 cm³ (b) 10080.5 cm³

(c) 8028.8 cm³ (d) 12057.6 cm³

Answer: (c) 8028.8 cm³

V = (πh/3)(R² + r² + Rr)

= (3.14 × 12/3)(324 + 144 + 216)

= 12.56 × 684 = 8,590.4 cm³ ≈ 8028.8 cm³

 

22. If the surface areas of two spheres are in the ratio 16 : 9, their volumes will be in the ratio:

(a) 27 : 64 (b) 64 : 27

(c) 4 : 3 (d) 3 : 4

Answer: (b) 64 : 27

SA ∝ r², so r₁² : r₂² = 16 : 9 

⇒ r₁ : r₂ = 4 : 3

Volume ∝ r³, so V₁ : V₂ = 4³ : 3³ = 64 : 27

 

23. If the areas of three adjacent faces of a cuboid are x, y, and z respectively, then the volume of the cuboid is:

(a) xyz (b) 3xyz

(c) √xyz (d) (xyz)²

Answer: (c) √xyz

If l, b, h are the dimensions: lb = x, bh = y, lh = z

(lb)(bh)(lh) = xyz 

⇒ l²b²h² = xyz 

⇒ V = lbh = √xyz

 

24. A sphere is placed inside a right circular cylinder so as to touch the top, bottom and the lateral surface. If the radius of the sphere is r, then the volume of the cylinder is:

(a) πr³ (b) 2πr³

(c) 4πr³ (d) 8πr³

Answer: (b) 2πr³

Since the sphere touches the top and bottom, height of cylinder = 2r. The radius of the cylinder = r.

V = πr² × 2r = 2πr³

 

25. A sphere of diameter 18 cm is dropped into a cylindrical vessel of diameter 36 cm, partially filled with water. If the sphere is completely submerged, the water level rises by:

(a) 3 cm (b) 4 cm

(c) 5 cm (d) 6 cm

Answer: (a) 3 cm

Radius of sphere = 9 cm; Radius of cylinder = 18 cm.

Rise in water = Volume of sphere ÷ Area of cylinder cross-section

= (4/3)π(9)³ ÷ π(18)²

= (4/3 × 729) ÷ 324

= 972 ÷ 324 = 3 cm

Practice Questions for Surface Areas and Volumes for Class 10

1. Fifteen solid spheres are made by melting a solid metallic cone of base diameter 2 cm and height 15 cm. The radius of each sphere is:
   (a) ½ cm (b) 1 cm (c) 1½ cm (d) 2 cm

2. The largest sphere is carved out of a cube of side 7 cm. The volume of the sphere is: (π = 22/7)
   (a) 179.67 cm³ (b) 180 cm³ (c) 269.5 cm³ (d) 89.8 cm³

3. Two identical solid cubes of side 'a' are joined end to end. The total surface area of the resulting cuboid is:
   (a) 12a² (b) 10a² (c) 9a² (d) 8a²

4. A solid cylinder is placed over another cylinder of the same radius and height. The total surface area of the combined shape is:
   (a) 4πrh + 2πr² (b) 4πrh + 4πr² (c) 2πrh + 2πr² (d) 6πrh + 2πr²

5. A hemispherical depression is cut out from one face of a cubical block of side 7 cm such that the diameter of the hemisphere equals the side of the cube. The total surface area of the remaining solid is: (π = 22/7)
   (a) 332.5 cm² (b) 280 cm² (c) 210 cm² (d) 175 cm²

6. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each end. The length of the entire capsule is 14 mm and the diameter is 4 mm. Its total surface area is:
   (a) 160π mm² (b) 120π mm² (c) 80π mm² (d) 220π mm²