Volume and Surface Area Aptitude

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.

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?

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?

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.

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.

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?

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?

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?

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.

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?

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.

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?

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?

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?

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?

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.

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.

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?

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.

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.

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