Surface Area of a Regular Pyramid

Surface Area of a Regular Pyramid

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

Pyramid

A pyramid is a 3D figure that has a vertex, a base, and lateral faces which are adjacent to the vertex and the base.

A pyramid is a 3D figure
that has
a [vertex], a [base] (brown),
and [lateral faces]
which are adjacent to the vertex and the base.

So the [lateral faces] are triangles.

Regular Pyramid

A regular pyramid is a pyramid whose base is a regular polygon. and whose lateral faces are all congruent.

A regular pyramid is a pyramid
whose base is a regular polygon
and whose lateral faces are all congruent.

So the vertex and the center of the base
are the endpoints of the height.

Formula

A = B + (1/2)*P*h_s. A: Surface area of a regular pyramid, B: Base area, P: Perimeter of the base, h_s: Slant height of the regular pyramid

A = B + (1/2)Phs

A: Surface area of a regular pyramid
B: Base area
P: Perimeter of the base
hs: Slant height of the regular pyramid
(= Height of the lateral face triangle)

(1/2)Phs: lateral area

Example

Find the surface area of the given regular pyramid. The side of the base: 10, Height: 12.

The base is a square.
Its side is 10.

So B = 102.

Area of a square

102 = 100

So B = 100.

The side of the base is 10.
And there are four congruent sides.

So P = 10⋅4

10⋅4 = 40

So P = 40.

Recall that,
for a regular pyramid,
the vertex and the center of the base
are the endpoints of the height.

So the intersection of the height and the base
is the center of the base.

So the right triangle's bottom leg is, 10/2, 5.

Then see this right triangle.

Starting from the shortest side,
the sides are (5, 12, hs).

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

Pythagorean triples

So hs = 13.

B = 100
P = 40
hs = 13

So A = 100 + (1/2)⋅40⋅13.

(1/2)⋅40 = 20

20⋅13 = 260

100 + 260 = 360

So A = 360.