Surface Area of a Right Prism

Surface Area of a Right Prism

How to find the surface area of a right prism: definition of a prism and its parts, formula, proof, examples, and their solutions.

Prism

This shape is called a prism.

This shape is called a prism.

A prism has two bases. The bases are polygonal faces that are congruent and parallel.

A prism has two bases.

The bases are polygonal faces
that are congruent and parallel.

So a prism is a 3D shape
that has a pair of bases.

The lateral faces of a prism are the faces that are not the bases.

The lateral faces of a prism are the faces
that are not the bases.

Right Prism

A right prism is a prism whose lateral faces are all rectangles.

A right prism is a prism
whose lateral faces are all rectangles.

Formula

A = 2B + Ph, A: surface area of a right prism, B: base area, P: perimeter of the base, h: height of the right prism

A = 2B + Ph

A: surface area of a right prism
B: base area
P: perimeter of the base
h: height of the prism

2B: sum of the base areas
Ph: lateral area

Proof

Surface Area of a Right Prism: Proof of the Formula

Draw the net of a right prism.

There are two bases.
So the sum of the base areas is 2B.

See the lateral faces in the net.
Each lateral face is a rectangle.
So the whole lateral faces form a rectangle.

Its width is the perimeter of the base: P.
And its height is h.

So the lateral area is Ph.

Area of a rectangle

So A = 2B + Ph.

Example 1

Find the surface area of the given right prism. Edges: 8, 7, 3.

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

B = 7⋅3 = 21

P = (7 + 3)⋅2 = 20

h = 8

B = 21, P = 20, h = 8

A = 2⋅21 + 20⋅8

Example 2

Find the surface area of the given right prism. Edges: 4, 7.

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

B = (√3/4)⋅42 = 4√3

Area of an equilateral triangle

P = 4⋅3 = 12

h = 7

B = 4√3, P = 12, h = 7

A = 2⋅4√3 + 12⋅7