## Properties of Pyramid

This tool calculates the basic geometric properties of a right pyramid, with a square base. Enter the shape dimensions 'a' and 'h' below. The calculated results will have the same units as your input. Please use consistent units for any input.

 a = h = Geometric properties: Volume = Surface area = Base area = Lateral surface area = Lateral edge length L =

## Definitions

### Geometry

The pyramid is a solid object with a polygonal base and side faces that converge to a single point, the apex. Typically the most recognizable pyramid is the one with square base. This type is examined in this page. A square based pyramid, has 5 faces, 8 edges and 5 vertices.

The volume of a square based pyramid is given by the formula:

$V = \frac{1}{3} a^2 h$

where a the length of a base side and h the height of the pyramid. This formula is derived from the following general expression, that covers all pyramids, with any polygonal base, and also the cone:

$V = \frac{1}{3} A_b h$

where $$A_b$$ the base area.

The surface area of one side face of the square based pyramid, can be found using the following formula:

$A_{f0} = \frac{a}{2}\sqrt{h^2 + \frac{a^2}{4}}$

Since there are four side faces and the area of the square base is equal to $$a^2$$, the total surface area of the square based pyramid is:

$A = a^2 + 4A_{f0}$

The edge, that connects a base vertex with apex, has a length equal to:

$L = \sqrt{h^2 + \frac{a^2}{2}}$