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]Phs

A: Surface area of a regular pyramid
B: Base area
P: Perimeter of the base
hs: 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 = 102.

102 = 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 hs.

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 hs.

Then the sides of the triangle are (5, 12, hs).

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

Pythagorean Triple

So the slant height, hs, is 13.

B = 100
P = 40
hs = 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.