Pyramids and cones: pointed solids

A common formula

A pyramid and a cone are solids that taper to a point (the apex). They share the same formula for volume:

V = (Area of the base × height) / 3

This formula shows that, for the same base and height, a pyramid (or a cone) has a volume three times smaller than that of a right-angled prism or a cylinder.

Volume of a pyramid

If the base is a square with side length c and the height of the pyramid is h:

V = (c² × h) / 3

Example: a pyramid with a square base of side length 6 cm and height 9 cm: V = (6² × 9) / 3 = (36 × 9) / 3 = 324 / 3 = 108 cm³

Volume of a cone

If the radius of the base is r and the height is h:

V = (π × r² × h) / 3

Example: a cone with radius r = 4 cm and height h = 9 cm: V = (π × 4² × 9) / 3 = (π × 16 × 9) / 3 = (144 × π) / 3 = 48 × π ≈ 150.8 cm³

A common pitfall to avoid

The height h used in the formula is the perpendicular height between the apex and the base, not the length of a lateral edge (sometimes called the ‘apothema’ or height of a face). These must not be confused, otherwise the calculation will be incorrect.