Prism questions are among the most reliable mark-earners at GCSE. The formula is straightforward — area of the cross-section multiplied by the length — but success depends on correctly identifying the cross-section and calculating its area first. This topic appears on both Foundation and Higher papers across AQA, Edexcel and OCR.
What Is a Prism?
A prism is a three-dimensional shape with a uniform cross-section that runs the entire length of the solid. If you sliced a prism at any point along its length, you would always get the same 2D shape.
Common prisms include cuboids (rectangular cross-section), triangular prisms, and shapes with L-shaped, trapezoidal, or pentagonal cross-sections. A cylinder is technically a circular prism, though it has its own formula.
The key idea is that every prism can be thought of as a 2D shape stretched along a length. This is why the volume equals the area of that 2D shape (the cross-section) multiplied by how far it is stretched (the length or depth).
Prism questions at GCSE range from straightforward triangular prisms on Foundation to prisms with compound cross-sections (such as L-shapes or trapeziums) on Higher papers. In every case, the method is the same — find the cross-section area first, then multiply by the length.
Key Formulas
Step-by-Step Method
- Identify the cross-section — the 2D shape that stays the same along the length of the prism.
- Calculate the area of the cross-section using the appropriate area formula (triangle, trapezium, L-shape, etc.).
- Multiply the cross-sectional area by the length (or depth) of the prism.
- State your answer with cubic units (cm³, m³, etc.).
Worked Example 1 — Foundation Level
Question: A triangular prism has a triangular cross-section with base 8 cm and perpendicular height 5 cm. The prism is 12 cm long. Find its volume.
Working: Cross-section area = ½ × 8 × 5 = 20 cm² Volume = 20 × 12 = 240
Answer: 240 cm³
Worked Example 2 — Higher Level
Question: A prism has a trapezoidal cross-section. The trapezium has parallel sides of 6 cm and 10 cm and a perpendicular height of 4 cm. The prism is 15 cm long. Find its volume.
Working: Cross-section area = ½(6 + 10) × 4 = ½ × 16 × 4 = 32 cm² Volume = 32 × 15 = 480
Answer: 480 cm³
Worked Example 3 — Exam Style
Question: A prism has an L-shaped cross-section. The L-shape has overall dimensions 7 cm by 5 cm, with a 3 cm by 2 cm rectangle removed from one corner. The prism is 20 cm long. Find the volume.
Working: Full rectangle area = 7 × 5 = 35 cm² Removed rectangle area = 3 × 2 = 6 cm² L-shaped cross-section area = 35 − 6 = 29 cm² Volume = 29 × 20 = 580
Answer: 580 cm³
Common Mistakes
- Confusing the cross-section with a side face. The cross-section is the face that stays the same along the length. It is usually the shape shown "face-on" in the diagram.
- Using the slant side of a triangle instead of the perpendicular height. When the cross-section is a triangle, you need the height at right angles to the base.
- Multiplying by the wrong dimension. After finding the cross-sectional area, multiply by the length that runs perpendicular to the cross-section, not another dimension of the cross-section.
- Forgetting cubic units. Volume is measured in cm³ or m³, not cm² or cm. If your units are wrong, the examiner knows you have confused area and volume.
Exam Tips
- Always write "area of cross-section = ..." as a separate step. This earns a method mark and keeps your working clear.
- If the cross-section is a compound shape, find its area using addition or subtraction of simpler shapes before multiplying by the length.
- The prism formula is not on the formula sheet for AQA or Edexcel — you must memorise it.
- When asked for volume in a real-world context (e.g. filling a container), check if you need to convert units or state capacity in litres (1000 cm³ = 1 litre).
Practice Questions
Q1 (Foundation): A triangular prism has a cross-section with base 6 cm and height 4 cm. The prism is 10 cm long. Find its volume.
Q2 (Foundation): A cuboid measures 9 cm by 4 cm by 5 cm. Find its volume.
Q3 (Higher): A prism has a pentagonal cross-section with area 42 cm². The prism is 18 cm long. Find its volume.
Practise volume of a prism questions with instant feedback — completely free on GCSEMathsAI.
Related Topics
Summary
- A prism has a uniform cross-section that runs along its entire length.
- Volume of a prism = area of cross-section × length.
- Always identify and calculate the cross-sectional area first, then multiply by the length.
- Common cross-sections include triangles, trapeziums, and L-shapes.
- This formula is not given on the exam formula sheet — learn it by heart.
- Show the cross-sectional area calculation as a clear separate step in your exam working.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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