Volume of Prisms
Every question with a fully worked solution.
- Volume of Prisms - Video - Volume of a Prism Watch
Theory
Every prism follows one rule: volume = area of the cross-section × length. This Year 9 maths page uses it for cubes, rectangular, triangular and trapezium prisms and solids with curved cross-sections. It also covers holes, turning a volume into a capacity in litres, and working backwards to a missing length.
A prism is a solid with the same shape all the way through. That shape is the cross-section. A cylinder is like a prism with a circle as its cross-section.
The volume is the amount of space inside a solid. It is measured in cubic units, such as \(\text{cm}^3\) or \(\text{m}^3\).
To find the volume, find the area of the cross-section. Then multiply by the length (or height) of the solid, measured at right angles to the cross-section.
A composite solid is made of two or more simple solids. Add their volumes, or subtract a part that has been cut out, such as a hole.
The capacity of a container is how much it holds. It is measured in millilitres (mL) and litres (L).
For any prism, with \(A\) the area of the cross-section and \(h\) the length:
The common cases are the same rule:
| Cross-section | Area \(A\) |
|---|---|
| Triangle | \(\dfrac{1}{2} \times \text{base} \times \text{height}\) |
| Parallelogram | \(\text{base} \times \text{height}\) |
| Rhombus or kite | \(\dfrac{1}{2} \times\) the product of the diagonals |
| Trapezium | \(\dfrac{h}{2}(a + b)\), with \(a\) and \(b\) the parallel sides |
| Circle | \(\pi r^2\) |
| Semicircle | \(\dfrac{1}{2}\pi r^2\) |
| Quadrant | \(\dfrac{1}{4}\pi r^2\) |
How to find the volume of a prism
- Find the cross-section: the face that is the same all the way through. It may be at the front, the end or the top of the diagram.
- Work out the area of the cross-section, \(A\).
- Multiply by the length: \(V = A \times h\). For a solid with a hole, subtract the volume of the hole.
- Finish with cubic units, round as asked, or convert to mL or L. To find a missing length, put the volume in and solve.
Find the area of the triangle first. It is the face that repeats.
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 7 \times 4\) |
| \(\) | \(=\) | \(14 \text{ cm}^2\) |
Multiply by the length of the prism:
| \(V\) | \(=\) | \(14 \times 9\) |
| \(\) | \(=\) | \(\mathbf{126} \textbf{ cm}^\mathbf{3}\) |
The \(9\) cm is not part of the triangle. It is how far the triangle runs.
(i) its volume
(ii) how many litres it holds when full.
(i) The trapezium is the face that repeats. Its parallel sides are \(0.6\) m and \(0.4\) m, and its height is the depth.
| \(A\) | \(=\) | \(\dfrac{0.5}{2} \times (0.6 + 0.4) = 0.25 \text{ m}^2\) |
| \(V\) | \(=\) | \(0.25 \times 2 = \mathbf{0.5} \textbf{ m}^\mathbf{3}\) |
(ii) \(1 \text{ m}^3 = 1000\) L.
| \(0.5 \times 1000\) | \(=\) | \(\mathbf{500} \textbf{ L}\) |
The area of the triangle needs both shorter sides. Find the missing one with Pythagoras.
| \(b^2\) | \(=\) | \(17^2 - 8^2 = 289 - 64 = 225\) |
| \(b\) | \(=\) | \(15 \text{ cm}\) |
The two shorter sides are perpendicular, so they are the base and height of the triangle.
| \(A\) | \(=\) | \(\dfrac{1}{2} \times 8 \times 15 = 60 \text{ cm}^2\) |
| \(V\) | \(=\) | \(60 \times 12 = \mathbf{720} \textbf{ cm}^\mathbf{3}\) |
Using \(17\) as the height would be wrong: the hypotenuse is not perpendicular to either short side.
(ii) A cube has a volume of \(343 \text{ cm}^3\). How long is each edge?
Write the formula, put in what you know, then solve for the missing length.
(i)
| \(1800\) | \(=\) | \(20 \times 9 \times h\) |
| \(1800\) | \(=\) | \(180h\) |
| \(h\) | \(=\) | \(1800 \div 180 = \mathbf{10} \textbf{ cm}\) |
(ii) \(V = s^3\), so undo the cube with a cube root.
| \(s^3\) | \(=\) | \(343\) |
| \(s\) | \(=\) | \(\sqrt[3]{343} = \mathbf{7} \textbf{ cm}\) |
Check: \(7 \times 7 \times 7 = 343\) \(\checkmark\)
The same steps work when the sides are written with a pronumeral. Sides of \(2x\), \(x\) and \(4x\) cm with a volume of \(1000 \text{ cm}^3\) give \(8x^3 = 1000\), so \(x^3 = 125\) and \(x = \sqrt[3]{125} = 5\).
Common pitfalls
Frequently asked questions
What is the formula for the volume of a prism?
\(V = A \times h\): the area of the cross-section times the length of the prism. The cross-section is the face that is the same all the way through.
How do you find the volume of a triangular prism?
Find the area of the triangle, \(\dfrac{1}{2} \times \text{base} \times \text{height}\), then multiply by the length of the prism.
How do you convert cubic centimetres to millilitres?
\(1 \text{ cm}^3 = 1 \text{ mL}\), so the number stays the same. \(1 \text{ m}^3 = 1000 \text{ L}\).
How do you find the height of a prism from its volume?
Divide the volume by the area of the cross-section. A rectangular prism with volume \(1800 \text{ cm}^3\) and base \(20\) cm by \(9\) cm has height \(1800 \div 180 = 10\) cm.