# Surface Area of a Regular Pyramid

How to find the surface area of a regular pyramid: definition of a pyramid and its parts, formula, proof, examples, and its solution.

## Pyramid

This shape is called a pyramid.

A pyramid has one base and a vertex.

The height is the distance

between the base's plane and the vertex.

The lateral faces of a prism are the faces

that are not the base.

## Regular Pyramid

A regular pyramid is a pyramid

whose base is a regular polygon.

The height of a regular pyramid

meets with the center of the base.

The slant height of a regular pyramid

is the height of the lateral face.

## Formula

*A* = *B* + (1/2)*Ph _{s}*

*A*: surface area of a regular pyramid

*B*: base area

*P*: perimeter of the base

*h*: slant height of the regular pyramid

_{s}(1/2)

*Ph*: lateral area

_{s}## Proof

Draw the net of a regular pyramid.

Set the side of the base as *a*.

The base area is *B*.

See the lateral faces in the net.

Each lateral face is a triangle

whose base is *a*.

For an *n*-gon base,

there are *n* of those triangles.

So *P* = *a*⋅*n*.

The lateral area is formed by *n* of triangles.

So the lateral area is (1/2)*ah _{s}*⋅

*n*.

Area of a triangle

Then

*A*=

*B*+ (1/2)

*ah*⋅

_{s}*n*.

*P*=

*a*⋅

*n*

So

*A*=

*B*+ (1/2)

*Ph*.

_{s}## Example

To use the formula,

find *B*, *P* and *h _{s}*.

To find

*h*,

_{s}draw a right triangle

that includes the height and the slant height.

It's legs are 5 (= 10/2) and 12.

So this is a (5, 12, 13) triangle.

So

*h*= 13.

_{s}*B* = 10^{2} = 100

Area of a square*P* = 10⋅4 = 40

*B* = 100, *P* = 40, *h _{s}* = 13

*A*= 100 + (1/2)⋅40⋅13