Volume of a Pyramid

Calculates the volume of a pyramid as base area × height ÷ 3, whatever the shape of the base. It is one third of the prism with the same base and height.

Explanation

A pyramid joins a polygonal base to a single apex above it. Whether the base is a triangle, a square or any other polygon, the volume depends on nothing but the base area and the height.

V = \dfrac{1}{3} S h

$V$ — the volume
$S$ — the area of the base
$h$ — the height, the perpendicular distance from the base to the apex

Example

With the defaults, a base area of $S = 25$ and a height of $h = 9$ — for instance a square pyramid whose base is 5 by 5 and which stands 9 tall.

V = \dfrac{1}{3} \times 25 \times 9 = 75

The volume is 75. A prism with the same base and height would hold $25 \times 9 = 225$ , so the pyramid is exactly one third of it.

Watch out

The input is the area of the base, not the length of a side. For a square base of side 5, enter $5 \times 5 = 25$ , not 5. The height is the perpendicular drop from the apex to the plane of the base; it is neither a slanting edge nor the slant height of a triangular face. Note also that the apex need not sit directly above the base: an oblique pyramid with the same base area and the same height has exactly the same volume. Base area and height must both be positive.