Volume of a Prism
Theory
A prism has the same cross-section all the way through. Its volume is the area of the cross-section times the height: \(V = Ah\). A triangular prism with a right-angled cross-section of sides \(4\) cm and \(5\) cm, \(6\) cm long, has \(V = 10 \times 6 = 60~\text{cm}^3\). This Year 8 maths page shows how.
The shape that runs all the way through is the cross-section. Every slice parallel to the ends is the same shape and size: a uniform cross-section.
\(V = Ah\), where \(A\) is the area of the cross-section and \(h\) is the distance between the ends, measured at right angles to the cross-section. When a prism lies on its side, \(h\) is its length.
Two steps, every time: find \(A\), then multiply by \(h\).
Areas you need: rectangle \(lw\); triangle \(\dfrac{1}{2}bh\); parallelogram \(bh\); trapezium \(\dfrac{1}{2}(a + b)h\); rhombus or kite \(\dfrac{1}{2}xy\); sector \(\dfrac{\theta}{360} \times \pi r^2\).
A hole straight through belongs to the cross-section: take its area off \(A\) first.
Working backwards: \(A = V \div h\) or \(h = V \div A\). A cube has \(V = s^3\), so \(s = \sqrt[3]{V}\).
Scaling and context. Halving the area halves the volume; tripling the length triples it. Cost \(=\) volume \(\times\) price per cubic metre; \(1~\text{m}^3\) of water is \(1\) kL.
Volume of a prism:
How to find the volume of a prism
- Identify the cross-section and the height \(h\) at right angles to it.
- Find the area \(A\) of the cross-section.
- Multiply: \(V = Ah\), in cubic units.
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 4 \times 5 = 10\) |
| \(V\) | \(=\) | \(10 \times 6 = 60\) |
\(60~\text{cm}^3\)
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 3 \times (2 + 5) = \dfrac{21}{2}\) |
| \(V\) | \(=\) | \(\dfrac{21}{2} \times 4 = 42\) |
\(42~\text{cm}^3\) (cubic units, because it is a volume).
(i) the area of the cross-section
(ii) the volume
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 8 \times 7 = 28\) |
| \(V\) | \(=\) | \(28 \times 5 = 140\) |
(i) \(28~\text{cm}^2\)
(ii) \(140~\text{cm}^3\)
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 0.5 \times 0.25\) |
| \(\) | \(=\) | \(0.0625\) |
| \(V\) | \(=\) | \(0.0625 \times 8 = 0.5\) |
| \(0.5 \times 220\) | \(=\) | \(110\) |
The concrete costs \(\$110\).
(ii) A cube has a volume of \(512~\text{cm}^3\). How long is each edge?
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 9 \times 12 = 54\) |
| \(h\) | \(=\) | \(486 \div 54 = 9\) |
| \(s\) | \(=\) | \(\sqrt[3]{512} = 8\) |
(i) \(9\) cm
(ii) \(8\) cm, because \(8 \times 8 \times 8 = 512\).
| \(A\) | \(=\) | \(\dfrac{60}{360} \times \pi \times 15^2 = 37.5\pi\) |
| \(V\) | \(=\) | \(37.5\pi \times 8 = 300\pi\) |
| \(\) | \(=\) | \(942.47\ldots\) |
\(942.5~\text{cm}^3\)
Common pitfalls
Frequently asked questions
What is a prism?
A solid with the same cross-section all the way through, such as a triangular prism.
What is the formula for the volume of a prism?
\(V = Ah\): area of the cross-section times the height.
What is a cross-section?
The shape you see when you slice a solid parallel to its ends.
How do I find the length of a prism from its volume?
Divide the volume by the area of the cross-section: \(h = V \div A\).
How do I find the edge of a cube from its volume?
Take the cube root: \(s = \sqrt[3]{V}\).