Volume of a Prism
Volume of a prism is the amount of 3D space a prism takes up, found by multiplying the area of its cross-section by its length. It is a core GCSE Geometry and Measures topic covering cuboids, triangular and trapezium-based prisms and compound solids, with answers usually given in cubic units such as cm^3 or m^3.
Before you start
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Method
- Identify the shape of the cross-section, the face that stays the same all along the prism.
- Work out the area of the cross-section using the correct 2D formula.
- Multiply the cross-sectional area by the length of the prism.
- Give the units as the cube of the length units used (cm^3, m^3, and so on).
- For compound prisms, split the cross-section into simple shapes, find each area, then add or subtract before multiplying by length.
- If the volume is already given, rearrange (length = volume divided by cross-sectional area) to find a missing dimension.
Worked example
A prism has a cross-section that is a right-angled triangle with base 9 cm and height 6 cm. The prism is 12 cm long. Work out the volume of the prism.
- Area of the triangular cross-section = 0.5 x 9 x 6 = 27 cm^2.
- Volume = cross-sectional area x length.
- Volume = 27 x 12.
- Final answer: 324 cm^3.
Practice questions
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Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
A cuboid has a square cross-section with side length x cm and length 12 cm. The volume of the cuboid is 432 cm^3. Work out the value of x.
A prism has a cross-section that is a right-angled triangle. The hypotenuse of the triangle is 17 cm and one of the other two sides is 8 cm. The prism is 11 cm long. Work out the volume of the prism.
A prism has a cross-section that is a right-angled triangle. The base of the triangle is x cm and the height of the triangle is (x + 5) cm. The prism is 6 cm long. The volume of the prism is 252 cm^3. (a) Show that x^2 + 5x - 84 = 0. (b) Hence work out the value of x, given that x > 0.
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