Volume

Volume is how much space a solid takes up, measured in cubic units such as cm³ and m³. Count the cubes, or multiply: for a cuboid, length × width × height; for any prism or cylinder, the area of the cross-section × the length.

Key facts

  • Volume is measured in cubic units: count the 1 cm cubes that fill a solid.
  • Cuboid: V = length × width × height.
  • Any prism: V = area of cross-section × length.
  • Cylinder: V = πr²h.
  • 1 litre = 1,000 cm³, and 1 cm³ holds 1 ml.

Worked examples

Example 1

Find the volume of a cube with edges of 4 cm.

  1. Multiply the three edges. 4 × 4 × 4 = 64 cm³

Answer: 64 cm³

Example 2

A cuboid is 5 cm long, 3 cm wide and 2 cm high. Find its volume.

  1. Multiply the three lengths. 5 × 3 × 2 = 30 cm³

Answer: 30 cm³

Example 3

A box is filled with 1 cm cubes: 5 along, 3 deep and 2 layers high. How many cubes fill it, and what is its volume?

  1. One layer has 5 rows of 3 cubes. 5 × 3 = 15
  2. There are 2 layers. 15 × 2 = 30 cubes = 30 cm³

Answer: 30 cubes, 30 cm³

Example 4

A fish tank is 60 cm long, 30 cm wide and 40 cm tall. A tap fills it at 6 litres per minute. How long does it take to fill?

  1. Volume of the tank: 60 × 30 × 40 = 72,000 cm³
  2. 1 litre = 1,000 cm³, so divide by 1,000. 72,000 ÷ 1,000 = 72 litres
  3. The tap gives 6 litres every minute. 72 ÷ 6 = 12 minutes

Answer: 12 minutes

Example 5

A cuboid has a volume of 60 cm³. It is 5 cm long and 4 cm wide. How tall is it?

  1. Volume = length × width × height 5 × 4 × h = 60
  2. Divide. 60 ÷ 20 = 3 cm

Answer: 3 cm

Example 6

A prism has a cross-section with an area of 15 cm² and is 8 cm long. Find its volume.

  1. Volume of a prism = area of cross-section × length. 15 × 8 = 120 cm³

Answer: 120 cm³

Example 7

A triangular prism is 10 cm long. Its end is a triangle with base 6 cm and height 4 cm. Find its volume.

  1. Find the area of the triangle. ½ × 6 × 4 = 12 cm²
  2. Multiply by the length. 12 × 10 = 120 cm³

Answer: 120 cm³

Example 8

A cylinder has radius 3 cm and height 10 cm. Find its volume, to 1 d.p.

  1. The cross-section is a circle, with area πr². π × 3² = 9π cm²
  2. The same circle is stacked all the way down, 10 cm. 9π × 10 = 90π
  3. Use π on a calculator and round. 90π = 282.74… ≈ 282.7 cm³

Answer: 282.7 cm³

Example 9

A cylinder has a volume of 90π cm³ and a radius of 3 cm. Find its height.

  1. Substitute the radius. π × 3² × h = 90π
  2. Divide by 9π. h = 90 ÷ 9 = 10 cm

Answer: 10 cm

Example 10

Find the volume of a sphere with radius 3 cm, to 1 decimal place.

  1. Cube the radius, multiply by π, then by ⁴⁄₃. ⁴⁄₃ × π × 3³ = 36π = 113.097…
  2. Round to 1 decimal place. 113.1 cm³

Answer: 113.1 cm³