The volume of a pyramid is a Higher tier topic that builds on your knowledge of area and 3D shapes. The key idea is that a pyramid occupies exactly one-third the volume of a prism with the same base and height.
What Is a Pyramid?
A pyramid is a 3D shape with a flat polygon base and triangular faces that meet at a single point called the apex. The most common types in GCSE questions are square-based pyramids and triangular-based pyramids (tetrahedra).
The volume formula V = ⅓ × base area × height works for every pyramid, regardless of the shape of the base. The height used must be the perpendicular (vertical) height from the base to the apex, not the slant height along a triangular face.
Understanding why the factor is ⅓ is helpful: three identical pyramids can be assembled to form a prism with the same base and height. Each pyramid therefore contains one-third of that prism's volume.
Key Formulas
Step-by-Step Method
- Calculate the area of the base — use the appropriate area formula for the base shape (square, rectangle, triangle, etc.).
- Identify the perpendicular height of the pyramid — the vertical distance from the centre of the base straight up to the apex.
- Multiply the base area by the height, then divide by 3 (or multiply by ⅓).
Worked Example 1 — Foundation Level
Question: A square-based pyramid has a base with side length 6 cm and a perpendicular height of 10 cm. Find its volume.
Working: Base area = 6² = 36 cm² V = ⅓ × 36 × 10 V = ⅓ × 360 V = 120
Answer: 120 cm³
Worked Example 2 — Higher Level
Question: A pyramid has a rectangular base measuring 8 cm by 5 cm. Its volume is 80 cm³. Find the perpendicular height.
Working: Base area = 8 × 5 = 40 cm² V = ⅓ × base area × h 80 = ⅓ × 40 × h 80 = (40h)/3 240 = 40h h = 240 ÷ 40 h = 6
Answer: 6 cm
Worked Example 3 — Exam Style
Question: A triangular-based pyramid has a base that is an equilateral triangle with side 10 cm and perpendicular height of the triangle 8.66 cm. The pyramid's vertical height is 12 cm. Find its volume to the nearest whole number.
Working: Base area = ½ × 10 × 8.66 = 43.3 cm² V = ⅓ × 43.3 × 12 V = ⅓ × 519.6 V = 173.2 V ≈ 173
Answer: 173 cm³
Common Mistakes
- Using slant height instead of perpendicular height. The slant height runs along a triangular face. The formula requires the vertical height from base to apex. Use Pythagoras to convert if needed.
- Forgetting to divide by 3. Without the ⅓ factor, you calculate the volume of a prism, not a pyramid.
- Incorrect base area. For a triangular-based pyramid, the base is a triangle, not a square — use the triangle area formula for the base.
- Mixing up volume and surface area. Volume is in cubic units (cm³), surface area in square units (cm²).
Exam Tips
- The formula V = ⅓ × base area × height is given on most exam formula sheets — check your board.
- Always show the base area calculation as a separate step for clarity and to earn method marks.
- If the question gives the slant height, you will likely need Pythagoras' theorem to find the perpendicular height first.
Practice Questions
Q1 (Foundation): A square-based pyramid has base side 9 cm and perpendicular height 15 cm. Find its volume.
Q2 (Foundation): A rectangular-based pyramid has a base of 12 cm by 4 cm and a height of 9 cm. Find its volume.
Q3 (Higher): A square-based pyramid has volume 200 cm³ and base side 10 cm. Find the perpendicular height.
Practise volume of a pyramid questions with instant feedback — completely free on GCSEMathsAI.
Related Topics
Summary
- A pyramid has a polygon base and triangular faces meeting at an apex.
- Volume = ⅓ × base area × perpendicular height.
- The ⅓ factor is what distinguishes a pyramid from a prism.
- Always use the perpendicular height, not the slant height.
- Calculate the base area first as a separate step, then apply the formula.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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