Volume of rectangular prisms (cubic cm & m)
Theory
Volume is the space a solid takes up, measured in cubic units such as cubic centimetres (\(\text{cm}^3\)) and cubic metres (\(\text{m}^3\)). A rectangular prism is built from layers of cubes, so its volume is length \(\times\) width \(\times\) height. This Year 6 maths page shows how to find volumes, choose units, find a missing side, and link volume to capacity.
A cubic centimetre (\(1\,\text{cm}^3\)) is a cube \(1\) cm on each edge. A cubic metre (\(1\,\text{m}^3\)) is a cube \(1\) m on each edge.
The bottom layer of a prism has length \(\times\) width cubes. There are as many layers as the height, so V = length \(\times\) width \(\times\) height, or V = base area \(\times\) height.
A cube with edge \(5\) cm has volume \(5 \times 5 \times 5 = 125\,\text{cm}^3\).
To find a missing dimension, divide the volume by the area of the base: height \(= V \div (\text{length} \times \text{width})\).
Volume and capacity: \(1\,\text{cm}^3\) holds \(1\) mL, and \(1\,\text{m}^3\) holds \(1000\) L (\(1\) kL).
Volume of a rectangular prism:
Finding volume
- Check all three dimensions are in the same unit.
- Multiply length \(\times\) width (the base layer) \(\times\) height.
- Write cubic units.
Finding a missing dimension
- Multiply the two known dimensions.
- Divide the volume by that product.
(i) How many cubes are in one layer?
(ii) How many cubes are there altogether?
| \(5 \times 2\) | \(=\) | \(10\) |
| \(10 \times 3\) | \(=\) | \(30\) |
(i) \(10\)
(ii) \(30\) cubes, so \(30\,\text{cm}^3\)
| \(9 \times 4 \times 2\) | \(=\) | \(72\) |
\(72\,\text{cm}^3\).
| \(5 \times 5 \times 5\) | \(=\) | \(125\) |
\(125\,\text{cm}^3\).
| \(8 \times 4 \times 3\) | \(=\) | \(96\) |
\(96\,\text{m}^3\).
(ii) A prism has volume \(90\,\text{cm}^3\) and base area \(18\,\text{cm}^2\). How high is it?
| \(96 \div 24\) | \(=\) | \(4\) |
| \(90 \div 18\) | \(=\) | \(5\) |
(i) \(4\) cm
(ii) \(5\) cm
| \(10 \times 8 \times 5\) | \(=\) | \(400\) |
\(400\,\text{cm}^3\), which holds \(400\) mL.
(i) Find its volume.
(ii) How many litres does it hold?
| \(2 \times 1.5 \times 1\) | \(=\) | \(3\) |
| \(3 \times 1000\) | \(=\) | \(3000\) |
(i) \(3\,\text{m}^3\)
(ii) \(3000\) L
| \(6 \times 5 \times 3\) | \(=\) | \(90\) |
Volume needs multiplying, not adding: \(90\,\text{cm}^3\).
Common pitfalls
Frequently asked questions
How do you find the volume of a rectangular prism?
Multiply length × width × height.
What is a cubic centimetre?
The space taken by a cube \(1\) cm on each edge, written \(1\,\text{cm}^3\).
How do you find the height from the volume?
Divide the volume by length × width.
How much does 1 cm³ hold?
\(1\) mL.
How many litres fit in 1 m³?
\(1000\) L.