The volume of any prism is found by multiplying the area of the cross-section (the identical face that runs along the full length) by the length: Volume = Area of cross-section × length. Once you can find the area of the end face, the volume calculation is a single multiplication.
What is a prism?
A prism is a 3D shape with two identical, parallel end faces (the cross-sections) connected by rectangular faces along its length. Every cross-section parallel to the ends is the same shape and size — this is the defining property of a prism.
Common prisms at KS3:
| Prism | Shape of cross-section |
|---|---|
| Cuboid | Rectangle |
| Triangular prism | Triangle |
| Trapezoidal prism | Trapezium |
| L-shaped prism | L-shape (two rectangles joined) |
| Cylinder | Circle (technically a prism) |
A cuboid is a prism with a rectangular cross-section; its volume formula V = l × w × h is the same as Area of rectangle × height = lw × h.
What is the volume formula for a prism?
$$V = A \times l$$
where A is the area of the cross-section and l is the length (or depth or height — the dimension perpendicular to the cross-section).
The key step is always finding the cross-sectional area first, using the appropriate 2D area formula.
Worked example 1: triangular prism
A triangular prism has a right-angled triangular cross-section with base 6 cm and perpendicular height 4 cm. The length of the prism is 10 cm.
- Area of the triangular cross-section:
A = ½ × 6 × 4 = 12 cm² - Volume:
V = 12 × 10 = **120 cm³** ✓
Always identify the cross-sectional face (the triangle here) before multiplying by the length.
Worked example 2: trapezoidal prism
A prism has a trapezoidal cross-section with parallel sides 5 cm and 9 cm, and a perpendicular height between them of 4 cm. The prism is 7 cm long.
- Area of the trapezium:
A = ½ × (5 + 9) × 4 = ½ × 14 × 4 = 28 cm² - Volume:
V = 28 × 7 = **196 cm³** ✓
Worked example 3: L-shaped prism
An L-shaped prism has a cross-section that can be split into two rectangles:
- Rectangle A: 8 cm wide, 3 cm tall
- Rectangle B: 4 cm wide, 5 cm tall (the lower extension)
The prism is 6 cm long.
- Area of the L-shaped cross-section:
Area A = 8 × 3 = 24 cm²Area B = 4 × 5 = 20 cm²- Total cross-sectional area:
24 + 20 = 44 cm²
- Volume:
V = 44 × 6 = **264 cm³** ✓
Alternative split: Sometimes an L-shape is more easily split differently (e.g. as one large rectangle minus a missing corner rectangle). Either approach gives the same cross-sectional area.
How do you find the length of a prism given its volume?
Rearrange the formula: l = V ÷ A.
Example: A triangular prism has a volume of 180 cm³. Its triangular cross-section has base 6 cm and height 5 cm. Find the length.
- Cross-sectional area:
A = ½ × 6 × 5 = 15 cm² - Length:
l = 180 ÷ 15 = **12 cm** ✓
What units should you use?
Always match units throughout the calculation:
- If all lengths are in centimetres, the volume will be in cm³.
- If all lengths are in metres, the volume will be in m³.
- Do not mix units within one calculation. Convert first if necessary.
| Lengths in | Volume in |
|---|---|
| mm | mm³ |
| cm | cm³ |
| m | m³ |
What mistakes should you avoid?
- Using the wrong face as the cross-section. The cross-section is the face that is identical at both ends — not any rectangular face along the side.
- Forgetting the ½ in the triangle formula. The cross-section area for a triangular prism is
½ × base × height. Omitting the half doubles the answer. - Adding an area to a length. Every step requires consistent dimensions: area × length = volume. You cannot add an area to another length or mix steps.
- Leaving the cross-section unsimplified. For L-shapes and other compound faces, always find the complete cross-sectional area before multiplying by the length.
Frequently asked questions
Is a cylinder a prism?
A cylinder behaves exactly like a prism — its cross-section is a circle and the formula is Volume = πr² × h (circle area × height). Technically, a cylinder is not a polygon-based prism because its cross-section is a curve, but the same method applies: find the cross-sectional area, multiply by the length.
How is the volume of a prism different from its surface area?
Volume measures the space inside the prism (in cubic units). Surface area measures the total area of all faces (in square units). For a triangular prism, the surface area includes two triangular ends plus three rectangular side faces; the volume only uses the triangular cross-section multiplied by the length.
What if the cross-section is not a standard shape?
Divide the cross-section into simpler shapes (rectangles, triangles, semicircles) whose areas you can calculate. Add all the individual areas together to find the total cross-sectional area, then multiply by the length of the prism.
Does it matter which dimension I call the "length"?
In principle, no — volume is the same regardless of orientation. However, you must be consistent: the dimension you call the "length" must be perpendicular to the face you call the "cross-section". Choosing any consistent pairing gives the same answer.
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