A pyramid is a polyhedron with a polygonal base and triangular faces that meet at a point called an apex which is not in the plane of the polygonal base.

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.

## Lateral Surface Area

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 L . S . A . = 1 2 p l 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

L . S . A . = 1 2 ( 24 ) ( 5 ) = 60 inches 2

## Total Surface Area

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 T . S . A . = 1 2 p l + B 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 . = 1 2 ( 64 ) ( 17 ) + 256 = 544 + 256 = 800 inches 2

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.

## Volume

The ** volume ** of a pyramid equals one-third the area of the base times the altitude (height) of the pyramid. ( V = 1 3 B h ) .

** Example 3: **

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

V = 1 3 B h V = 1 3 ( 10 ) 2 ( 12 ) = 400 cm 2

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