Volume of a pyramid

The volume of a pyramid is the amount of space it encloses. Below are a few examples of different types of pyramids.

Formula for the volume of a pyramid

The volume, V, of a pyramid is:

where B is the area of the base and h is the height.

The volume of a prism is Bh. The volume of a pyramid that has the same base and height as the prism it is inscribed in is exactly one-third the volume of the prism. This is true for any pyramid that can be inscribed in a prism as long as the base and height are the same.

Volume of a frustum pyramid

If a plane cuts through a pyramid that is parallel to its base, a frustum is created.

A plane parallel to its base intersects the pyramid above, forming a smaller pyramid above, and a frustum below.

The volume formula for the frustum of a pyramid is

where h is the height of the frustum and A1 and A2 are the areas of the bottom and top bases.


A frustum is created from a right square pyramid. The height of the frustum is 3, and the two bases have side lengths of 5 and 7 respectively. What is the volume of the frustum?

The areas of the square bases are:

A1 = 72 = 49

A2 = 52 = 25

Using the volume formula above:

V =
=74 + 35