Volume of Rectangular Prism
Every question with a fully worked solution.
- Volume of Rectangular Prism - Video - Calculating the Volume of Rectangular Prisms Watch
Theory
The volume of a solid is the amount of space it takes up, measured in cubic units. A rectangular prism holds \(l \times w\) unit cubes in each layer and has \(h\) layers, so \(V = l \times w \times h\). This Year 7 maths page shows counting in layers, the formula, base area times height and working backwards.
A cube with \(1\) cm edges has a volume of one cubic centimetre, \(1~\text{cm}^3\). The volume of a solid is the number of these unit cubes that fill it.
Count in layers. Find the number of cubes in the bottom layer, then multiply by the number of layers.
The formula. A rectangular prism \(l\) long and \(w\) wide has \(l \times w\) cubes in each layer and is \(h\) layers high, so \(V = l \times w \times h\).
Base area times height. \(l \times w\) is the area of the base, \(A\), so \(V = A \times h\) (written \(V = Ah\)). This works for any prism, because a prism has the same cross-section all the way up.
A cube has every edge the same length \(s\), so \(V = s \times s \times s\).
Working backwards. Substitute what you know, then divide: \(180 = 6 \times 5 \times h\) gives \(h = 180 \div 30 = 6\). For the edge of a cube, find the number that multiplies by itself three times to give \(V\).
Packing bigger cubes. Count how many fit along each edge, then multiply. A box \(12\) cm by \(8\) cm by \(6\) cm holds \(6 \times 4 \times 3 = 72\) cubes with \(2\) cm edges.
Formulas:
How to find the volume of a rectangular prism
- Change all three lengths to the same unit.
- Multiply length \(\times\) width \(\times\) height (or base area \(\times\) height).
- Write the answer in cubic units.
How to find a missing length
- Substitute into \(V = l \times w \times h\).
- Multiply the known lengths.
- Divide the volume by that product.
| \(5 \times 2\) | \(=\) | \(10\) |
| \(10 \times 3\) | \(=\) | \(30\) |
\(30~\text{cm}^3\)
| \(9 \times 4 \times 2.5\) | \(=\) | \(36 \times 2.5\) |
| \(\) | \(=\) | \(90\) |
\(90~\text{cm}^3\)
(i) Find the volume of sand in m\(^3\).
(ii) Find the cost of the sand.
| \(30 \div 100\) | \(=\) | \(0.3\) |
| \(2 \times 1.5 \times 0.3\) | \(=\) | \(0.9\) |
| \(0.9 \times 60\) | \(=\) | \(54\) |
(i) \(0.9~\text{m}^3\)
(ii) \(\$54\)
(ii) A cube has a volume of \(512~\text{cm}^3\). Find the length of one edge.
| \(20 \times 15\) | \(=\) | \(300\) |
| \(4200 \div 300\) | \(=\) | \(14\) |
| \(8 \times 8 \times 8\) | \(=\) | \(512\) |
(i) \(14\) cm (check: \(20 \times 15 \times 14 = 4200\))
(ii) \(8\) cm (\(9 \times 9 \times 9 = 729\) is too big)
| \(35 \times 6\) | \(=\) | \(210\) |
\(210~\text{cm}^3\)
| \(12 \div 3\) | \(=\) | \(4\) |
| \(9 \div 3\) | \(=\) | \(3\) |
| \(6 \div 3\) | \(=\) | \(2\) |
| \(4 \times 3 \times 2\) | \(=\) | \(24\) |
\(24\) cubes. (Not \(648 \div 3\): each cube is \(27~\text{cm}^3\), and \(648 \div 27 = 24\).)
| \(0.8 \times 100\) | \(=\) | \(80\) |
| \(80 \times 45 \times 30\) | \(=\) | \(108\,000\) |
\(108\,000~\text{cm}^3\)
Common pitfalls
Frequently asked questions
What is the formula for the volume of a rectangular prism?
\(V = l \times w \times h\).
What does V = Ah mean?
Volume equals the area of the base times the height.
How do you find the volume of a cube?
\(V = s \times s \times s\).
How do you find the height from the volume?
Divide the volume by length times width.
What happens to the volume if every edge doubles?
It becomes \(8\) times bigger.