# Regular Pyramid: Surface Area

How to find the surface area of a regular pyramid: definition, formula, 1 example, and its solution.

## Pyramid

### Definition

A pyramid is a 3D figure

that has a vertex, a polygon base,

and triangle lateral faces.

## Regular Pyramid

### Definition

A regular pyramid is a pyramid

whose base is a regular polygon

and whose lateral faces are

all congruent isosceles triangles.

So the vertex and the center of the base

are the endpoints of the height.

## Formula

### Formula

A = B + [1/2]Ph_{s}

A: Surface area of a regular pyramid

B: Base area

P: Perimeter of the base

h_{s}: Slant height (= Height of the lateral face)

## Example

### Example

### Solution

Find the base area B.

The base is a square.

Its sides are all 10.

So the area of the square is

B = 10^{2}.

10^{2} = 100

So the base area B is 100.

Find the perimeter of the base P.

The base is a square.

Its sides are all 10.

There are 4 sides.

So the perimeter is

P = 4⋅10.

4⋅10 = 40

So the perimeter of the base P is 40.

Next, find the slant height h_{s}.

The base is a square.

Its side is 10.

So the distance between

the center of the base

and the side

is, 10/2, 5.

See this right triangle.

The height is 12.

Set the slant height h_{s}.

Then the sides of the triangle are (5, 12, h_{s}).

So this right triangle is

a (5, 12, 13) right triangle.

Pythagorean Triple

So the slant height, h_{s}, is 13.

B = 100

P = 40

h_{s} = 13

Then the surface area, A,

is equal to,

the base area B, 100

plus,

the lateral area, [1/2]⋅40⋅13.

So A = 100 + [1/2]⋅40⋅13.

[1/2]⋅40 = 20

+20⋅13 = 260

100 + 260 = 360

So the surface area of the regular pyramid is

360.