Formula sheet
Volume of pyramid
(1/3) × base area × height is given on the exam formula sheet. You must still find the base area yourself, use the vertical height (not a slant edge), and keep the 1/3.
What the symbols mean
- A Area of the base, any flat shape such as a square, rectangle or triangle (area unit).
- h Vertical height from the base up to the apex (length unit), not a slant edge.
Given in the exam, or memorise?
(1/3) × base area × height is given on the exam formula sheet. You must still find the base area yourself, use the vertical height (not a slant edge), and keep the 1/3.
Why it works
A pyramid is one-third of a prism on the same base.
- Take a prism with base area A and height h, volume A × h.
- A pyramid with the same base and height fills exactly one-third of it.
- So volume of pyramid = (1/3) × A × h.
Worked example 1
A pyramid has a square base of side 6 cm and vertical height 10 cm. Find its volume.
- Base area A = 6 × 6 = 36
- Volume = (1/3) × A × h
- = (1/3) × 36 × 10
- = 12 × 10 = 120
Worked example 2
A pyramid has volume 96 cm³ and base area 24 cm². Find its vertical height.
- Volume = (1/3) × A × h
- 96 = (1/3) × 24 × h
- 96 = 8h
- h = 96 ÷ 8 = 12
Where students go wrong
- Forgetting the 1/3 and getting the prism volume instead.
- Using a slant edge or slant height in place of the vertical height h.
- Computing the base area wrongly, especially for triangular or rectangular bases.
- Answering in cm² instead of cm³.
Use it with
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
What shape can the base of a pyramid be?
Any flat polygon, a square, rectangle, triangle, or other shape. The formula does not care about the base shape; you just work out its area A and put it in.
So first identify the base, calculate its area with the right area rule, then multiply by height and one-third.
Which height goes into the formula?
The vertical height, the perpendicular distance from the base straight up to the apex. Do not use a slant edge or the slant height of a face, which are longer.
If only a slant measurement is given, use Pythagoras with the base to find the vertical height before substituting into (1/3) × A × h.
Why is a pyramid one-third of a prism?
For the same base and the same height, three identical pyramids can be fitted together to fill one prism exactly. That is why the pyramid's volume carries the factor 1/3.
If you drop the 1/3, you compute the prism instead and your answer is three times too big.