Surface Area of a Regular Pyramid

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.

This shape is called a pyramid.

A pyramid has one base and a vertex. And the height is the distance between the base's plane and the vertex.

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.

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.

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 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.

The slant height of a regular pyramid
is the height of the lateral face.

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

(1/2)Phs: lateral area

Proof

Surface Area of a Regular Pyramid: Proof of the Formula

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 = an.

The lateral area is formed by n of triangles.
So the lateral area is (1/2)ahsn.

Area of a triangle

Then A = B + (1/2)ahsn.

P = an

So A = B + (1/2)Phs.

Example

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

To use the formula,
find B, P and hs.

To find hs,
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 hs = 13.

B = 102 = 100

Area of a square

P = 10⋅4 = 40

B = 100, P = 40, hs = 13

A = 100 + (1/2)⋅40⋅13