Sheet № 168 · Higher only · AQA · Edexcel · OCR
Volume of a Pyramid –
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.
§Key definitions
Question:
A square-based pyramid has a base with side length 6 cm and a perpendicular height of 10 cm. Find its volume.
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.
§Formulas to memorise
V = ⅓ × base area × perpendicular height
For a square-based pyramid with base side s: base area = s², so V = ⅓ × s² × h
Base area = 6² = 36 cm²
V = ⅓ × 36 × 10
V = ⅓ × 360
V = 120
Base area = 8 × 5 = 40 cm²
V = ⅓ × base area × h
h = 240 ÷ 40
h = 6
Worked example
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
⚠ 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.