Worksheet on Surface Areas and Volumes for Class 10

Solved Questions on Surface Area and Volume

1. The radius of the sphere, cone and cylinder is equal. Their total surface area is also equal. What is the ratio of their heights?

a) 1 : √2 : 1
b) 2 : 2 : 1
c) 2 : √2 : 1
d) 2 : 2√2 : 1

Answer: d) 2 : 2√2 : 1

Explanation: Let the radius of the sphere, cone and cylinder be R.

Let the height of the sphere be H₁.

Let the height of the cone be H₂.

Let the height of the cylinder be H₃.

Let l be the slant height of the cone.

We know that the height of the sphere is the same as the diameter of the sphere.

H₁ = 2R

We know that

Total Surface Area of the sphere = 4πR²

Total Surface Area of the cone = πR(l + R)

Total Surface Area of the cylinder = 2πR(H₃ + R)

We are given that the total surface area of the sphere, cone and cylinder is equal.

So first taking,

Total surface area of the sphere = Total surface area of the cylinder

4πR² = 2πR(H₃ + R)

2R = H₃ + R

R = H₃

Now taking,

Total surface area of the sphere = Total surface area of the cone

4πR² = πR(l + R)

4R = l + R

3R = l … (1)

In a cone,

(Slant height)² = Height² + Radius²

(l)² = (H₂)² + (R)²

(3R)² = (H₂)² + R²

9R² = (H₂)² + R²

9R² − R² = (H₂)²

8R² = (H₂)²

H₂ = √8R²

H₂ = 2√2 R

Now, 

H₁ : H₂ : H₃ = 2R : 2√2 R : R
                    = 2 : 2√2 : 1

2. A conical cup is filled with ice cream. The ice cream forms a hemispherical shape on its open top. The height of the hemispherical part is 4.2 cm. The height of the cone is equal to the radius of the hemispherical part. What is the volume of the ice cream?

a) 232.748 cm³
b) 232.648 cm³
c) 232.848 cm³
d) 232.548 cm³

Answer: c) 232.848 cm³

Explanation: We know that the volume of a hemisphere = 2πr³ / 3

Volume of a cone = πr²h / 3

We are given the height of the hemispherical part = 4.2 cm

Height of the hemispherical part = Radius of the hemisphere

The radius of the hemisphere = 4.2 cm

We are given that the height of the cone is equal to the radius of the hemispherical part.

Height of the cone = 4.2 cm

Now,

Volume of the ice cream = Volume of the hemispherical part + Volume of the cone
                                    = 2πr³/3 + πr²h/3
                                    = [²⁄₃ × ²²⁄₇ (4.2)³] + [¹⁄₃ × ²²⁄₇ × (4.2)² × 4.2]
                                    = [²⁄₃ × ²²⁄₇ × 4.2 ×  4.2 ×  4.2] + [¹⁄₃ × ²²⁄₇ × (4.2)² × 4.2]
                                    = (2 × 22 × 1.4 × 0.6 × 4.2) + (22 × 1.4 × 0.6 × 4.2)
                                    = (155.232) + (77.616)
                                    = 232.848 cm³

3. If a solid metallic sphere of radius 9 cm is melted and recast into a right circular cone of base radius 4 cm, then what is the height of the cone?

a) 180.25 cm
b) 182.25 cm
c) 183.25 cm
d) 181.25 cm

Answer: b) 182.25 cm

Explanation: We know that the volume of a sphere = 4πr³/3

Volume of a cone = πr²h / 3

The radius of the metallic sphere = 9 cm

The radius of the cone = 4 cm

Now, Volume of the metallic sphere = 4π (9)³ / 3
= 4π (729) / 3
= 4π (243)
= 972π

Volume of the cone = π(4)²h / 3
= 16πh / 3

When one solid is melted and recast into another shape, its volume is the same.

→ Volume of the cone = Volume of the metallic sphere

→ 16πh / 3 = 972π

→ h = $\frac{\frac{\mathrm{972\pi \; \times \; 3}}{}}{\frac{\mathrm{16\pi }}{}}$

→ h = $\frac{\frac{\mathrm{243\; \times \; 3}}{}}{\frac{4}{}}$
→ h = 729 / 4
→ h = 182.25 cm

4. If the area of the base of the frustum of a right circular cone is 25π cm², the diameter of the circular upper surface is 6 cm and the slant height is 8 cm, then what will be the total surface area of the frustum?

a) 308 cm²
b) 307 cm²
c) 304 cm²
d) 306 cm²

Answer: a) 308 cm²

Explanation: Let the radius of the base of the frustum be R cm.

We are given that the area of the base of the frustum is 25π cm².

We know that the area of the base is πr²

πR² = 25π
R² = 25
R = √25
R = 5 cm

We are given the diameter of the circular upper surface = 6 cm

The radius of the circular upper surface (r) = 3 cm

We are given that the slant height (l) = 8 cm

We know that the total surface area of the frustum = πR² + πr² + πl(R + r)

→ Total surface area of frustum = ²²⁄₇ × (5)² + ²²⁄₇ × (3)² + ²²⁄₇ × 8 × (5 + 3)
= ²²⁄₇ × 25 + ²²⁄₇ × 9 + ²²⁄₇ × 8 × 8
= ²²⁄₇ × (25 + 9 + 64)
= ²²⁄₇ × 98
= 22 × 14
= 308 cm²

5. If a solid sphere of radius 7 cm is melted and moulded into small identical cubes of side length 2 cm, then how many such cubes can be formed from the sphere?

a) 178
b) 181
c) 180
d) 179

Answer: d) 179

Explanation: We know that the volume of a sphere = 4πr³ / 3

Volume of the cube = (side)³

Volume of the sphere = 4πr³ / 3
                                = 4π(7)³ / 3
                                = (4 × (22 / 7) × (7)³ / 3
                                = (4 × 22 × (7)² / 3
                                = 4312 / 3
                                = 1437.33 cm³

Volume of the cube = (side)³
                             = (2)³
                             = 8 cm³

When one solid is melted and recast into another shape, its volume is the same.

Thus, Number of cubes formed = (Volume of the sphere) / (Volume of the cube)
                                             = 1437.33 / 8
                                             = 179.67
                                              ~ 179

Thus, 179 identical cubes of side 2 cm can be formed from the solid sphere with a radius of 7 cm when melted and moulded.