# 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 A_{1} and A_{2} are the areas of the bottom and top bases.

Example:

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:

A_{1} = 7^{2} = 49

A_{2} = 5^{2} = 25

Using the volume formula above:

V = | |

= | |

= | 74 + 35 |

= | 109 |