Volume and Surface Area are fundamental concepts in geometry, crucial in various fields including engineering, architecture, and everyday problem-solving. Volume measures the capacity of a three-dimensional object, while surface area is the total area that the surface of the object occupies. Understanding these concepts is essential for calculating the space occupied by objects and the material required to cover them.

## 1. Find the volume of a cube with each side measuring 5 cm.

a) 100 cm³

b) 125 cm³

c) 150 cm³

d) 175 cm³

### Answer:

b) 125 cm³

### Explanation:

```
Volume of a cube = side³.
Here, side = 5 cm.
Volume = 5³ = 5 * 5 * 5 = 125 cm³.
```

## 2. The surface area of a sphere is 314 cm². What is its radius?

a) 5 cm

b) 6 cm

c) 7 cm

d) 8 cm

### Answer:

a) 5 cm

### Explanation:

```
Surface area of a sphere = 4πr².
314 = 4 * π * r².
r² = 314 / (4π) ≈ 25.
r ≈ 5 cm.
```

## 3. A cylindrical water tank has a diameter of 14 meters and a height of 10 meters. What is the volume of the tank?

a) 1232 m³

b) 1540 m³

c) 1760 m³

d) 2000 m³

### Answer:

b) 1540 m³

### Explanation:

```
Volume of a cylinder = πr²h.
Radius (r) = Diameter / 2 = 14/2 = 7 meters.
Volume = π * 7² * 10 ≈ 1540 m³.
```

## 4. Find the total surface area of a cuboid with length 10 cm, breadth 8 cm, and height 6 cm.

a) 256 cm²

b) 272 cm²

c) 296 cm²

d) 320 cm²

### Answer:

b) 272 cm²

### Explanation:

```
Total surface area of a cuboid = 2(lb + bh + hl).
= 2(10*8 + 8*6 + 6*10) = 2(80 + 48 + 60) = 2 * 188 = 376 cm².
```

## 5. The radius of the base of a cone is 7 cm and its height is 24 cm. Find the volume of the cone.

a) 1232 cm³

b) 1358 cm³

c) 1408 cm³

d) 1508 cm³

### Answer:

c) 1408 cm³

### Explanation:

```
Volume of a cone = (1/3)πr²h.
r = 7 cm, h = 24 cm.
Volume = (1/3) * π * 7² * 24 ≈ 1408 cm³.
```

## 6. A sphere has a diameter of 14 cm. What is its volume?

a) 1437 cm³

b) 1794 cm³

c) 2156 cm³

d) 2436 cm³

### Answer:

a) 1437 cm³

### Explanation:

```
Volume of a sphere = (4/3)πr³.
Radius (r) = Diameter / 2 = 14/2 = 7 cm.
Volume = (4/3) * π * 7³ ≈ 1437 cm³.
```

## 7. A cube has a total surface area of 216 cm². What is the length of each side of the cube?

a) 4 cm

b) 5 cm

c) 6 cm

d) 7 cm

### Answer:

c) 6 cm

### Explanation:

```
Total surface area of a cube = 6 * side².
216 = 6 * side².
side² = 216 / 6 = 36.
side = √36 = 6 cm.
```

## 8. A cylinder has a radius of 5 cm and a height of 8 cm. What is its total surface area?

a) 314 cm²

b) 407 cm²

c) 502 cm²

d) 654 cm²

### Answer:

b) 407 cm²

### Explanation:

```
Total surface area of a cylinder = 2πr(h + r).
r = 5 cm, h = 8 cm.
Total surface area = 2 * π * 5 * (8 + 5) ≈ 407 cm².
```

## 9. Find the volume of a rectangular prism with length 12 cm, width 8 cm, and height 6 cm.

a) 576 cm³

b) 672 cm³

c) 748 cm³

d) 864 cm³

### Answer:

a) 576 cm³

### Explanation:

```
Volume of a rectangular prism = length * width * height.
= 12 * 8 * 6 = 576 cm³.
```

## 10. The surface area of a cube is 150 cm². What is the volume of the cube?

a) 125 cm³

b) 216 cm³

c) 343 cm³

d) 512 cm³

### Answer:

a) 125 cm³

### Explanation:

```
Surface area of a cube = 6 * side².
150 = 6 * side².
side² = 150 / 6 = 25.
side = √25 = 5 cm.
Volume = side³ = 5³ = 125 cm³.
```

## 11. A cone has a height of 9 cm and a volume of 84 cm³. Find the radius of the base.

a) 2 cm

b) 3 cm

c) 4 cm

d) 5 cm

### Answer:

b) 3 cm

### Explanation:

```
Volume of a cone = (1/3)πr²h.
84 = (1/3) * π * r² * 9.
r² = (84 * 3) / (π * 9) ≈ 9.
r ≈ 3 cm.
```

## 12. A hemispherical bowl has a radius of 7 cm. What is its total surface area?

a) 308 cm²

b) 462 cm²

c) 616 cm²

d) 792 cm²

### Answer:

a) 308 cm²

### Explanation:

```
Total surface area of a hemisphere = 3πr².
r = 7 cm.
Total surface area = 3 * π * 7² ≈ 308 cm².
```

## 13. What is the volume of a cylinder with a radius of 4 cm and a height of 10 cm?

a) 160 cm³

b) 320 cm³

c) 502 cm³

d) 640 cm³

### Answer:

a) 160 cm³

### Explanation:

```
Volume of a cylinder = πr²h.
r = 4 cm, h = 10 cm.
Volume = π * 4² * 10 ≈ 160 cm³.
```

## 14. The radius of a spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. What is the ratio of the surface area of the balloon in the two cases?

a) 1:2

b) 1:3

c) 1:4

d) 2:1

### Answer:

c) 1:4

### Explanation:

```
Surface area of a sphere = 4πr².
Ratio = (4π * 7²) / (4π * 14²) = 49 / 196 = 1 / 4.
```

## 15. A cuboidal box with dimensions 10 cm, 20 cm, and 15 cm is filled with water. What is the volume of water in the box?

a) 1500 cm³

b) 2000 cm³

c) 2500 cm³

d) 3000 cm³

### Answer:

d) 3000 cm³

### Explanation:

```
Volume = length * width * height.
= 10 * 20 * 15 = 3000 cm³.
```

## 16. A right circular cylinder has a volume of 1540 cm³ and a height of 14 cm. Find the radius of its base.

a) 5 cm

b) 7 cm

c) 9 cm

d) 10 cm

### Answer:

b) 7 cm

### Explanation:

```
Volume of a cylinder = πr²h.
1540 = π * r² * 14.
r² = 1540 / (π * 14) ≈ 49.
r ≈ 7 cm.
```

## 17. Question: A sphere and a cube have the same surface area. If the radius of the sphere is 7 cm, find the edge length of the cube.

a) 14 cm

b) 12 cm

c) 10 cm

d) 8 cm

### Answer:

a) 14 cm

### Explanation:

```
Surface area of the sphere = 4πr² = 4 * π * 7².
Surface area of the cube = 6 * side².
Equating both surface areas: 4 * π * 7² = 6 * side².
side² = (4 * π * 49) / 6.
side ≈ 14 cm (approximating π ≈ 3.14).
```

## 18. Question: The volume of a right circular cone is 1232 cm³. If the height is 24 cm, what is the radius of the base?

a) 7 cm

b) 8 cm

c) 9 cm

d) 10 cm

### Answer:

a) 7 cm

### Explanation:

```
Volume of a cone = (1/3)πr²h.
1232 = (1/3) * π * r² * 24.
r² = (1232 * 3) / (π * 24) ≈ 49.
r ≈ 7 cm (approximating π ≈ 3.14).
```

## 19. Question: A rectangular solid has a volume of 180 cm³. If its length, width, and height are in the ratio 3:2:1, find the length of the solid.

a) 9 cm

b) 12 cm

c) 6 cm

d) 15 cm

### Answer:

c) 6 cm

### Explanation:

```
Let the length = 3x, width = 2x, height = x.
Volume = length * width * height = 180 cm³.
3x * 2x * x = 180.
6x³ = 180.
x³ = 30.
x ≈ 3.11 cm.
Length = 3x ≈ 3 * 3.11 ≈ 9.33 cm ≈ 9 cm.
```

## 20. Question: A cylindrical container with a radius of 7 cm is filled with water to a height of 12 cm. Find the volume of water in the container.

a) 1848 cm³

b) 1932 cm³

c) 2060 cm³

d) 2210 cm³

### Answer:

a) 1848 cm³

### Explanation:

```
Volume of the cylinder = πr²h.
Volume = π * 7² * 12 ≈ 1848 cm³ (approximating π ≈ 3.14).
```