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Geometry

Volume of a pyramid

One third of the base area times the height, for a base of any shape.

V=13Bh

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See it move

s = 6h = 10
  • Base area B = s² = 36
  • Volume = ⅓Bh = 120 (a third of the box with the same base and height)

What the symbols mean

V
the volume
B
the area of the base (for a square base of side s, B=s2)
h
the perpendicular height from the base to the apex

When to use it

Use it for any pyramid, whatever the shape of its base: square, rectangular, triangular (a tetrahedron) or hexagonal. Work out the base area with the right area formula first. A cone is the same formula with a circular base.

Worked example

A pyramid has a square base 6 m on each side, so B = 36 m², and a height of 10 m. What is its volume?

  1. Start from the formula.

    V=13⁢B⁢h

  2. Substitute B = 36, h = 10 and work it out.

    V=13·36·10
    =120

Answer V = 120 m³

Why it works

Three identical pyramids fit together into a cube (each with the cube's face as base and a corner as apex), so each is a third of base × height. Shearing the apex sideways does not change the volume.

Common mistakes

  • Using the slant height of a face instead of the perpendicular height.
  • Putting the side length in for B instead of the base area.
  • Forgetting the ⅓.