Given a polygonal region *R* in a plane *E* and a point *V* not in *E*, the **pyramid **with base *R* and vertex *V* is the union of all line segments
such that *N* is a point of *R*.

Most of the pyramids that are studied in high school are **regular pyramids**. These pyramids have the following characteristics:

- 1) The base is a regular polygon.
- 2) All lateral edges are congruent.
- 3) All lateral faces are congruent isosceles triangles.
- 4) The altitude meets the base at its center.

The altitude of a lateral face of a regular pyramid is the **slant height.** In a non-regular pyramid, slant height is not defined.

The **lateral **surface area of a regular pyramid is the sum of the areas of its lateral faces.

The general formula for the **lateral surface area** of a regular pyramid is where *p* represents the perimeter of the base and *l* the slant height.

**Example 1:**

Find the lateral surface area of a regular pyramid with a triangular base if each edge of the base measures 8 inches and the slant height is 5 inches.

The perimeter of the base is the sum of the sides.

*p* = 3(8) = 24 inches

The **total surface area of a regular pyramid **is the sum of the areas of its lateral faces and its base.
The general formula for the **total surface area **of a regular pyramid is where *p* represents the perimeter of the base, *l* the slant height and *B* the area of the base.

**Example 2:**

Find the total surface area of a regular pyramid with a square base if each edge of the base measures 16 inches, the slant height of a side is 17 inches and the altitude is 15 inches.

The perimeter of the base is 4*s* since it is a square.

*p* = 4(16) = 64 inches

The area of the base is *s*^{2}.

*B = *16^{2} = 256 inches^{2}

*T. S. A. = *

There is no formula for a surface area of a non-regular pyramid since slant height is not defined. To find the area, find the area of each face and the area of the base and add them.

The **volume** of a pyramid equals one-third the area of the base times the altitude (height) of the pyramid. .

**Example 3:**

Find the volume of a regular square pyramid with base sides 10 cm and altitude 12 cm.