# 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.

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.

## Right Prism

A right prism is a prism

whose lateral faces are all rectangles.

## Formula

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

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

## Proof

Draw the net of a right prism.

There are two bases.

So the sum of the base areas is 2*B*.

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* = 2*B* + *Ph*.

## Example 1

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

To use the formula,

find *B*, *P*, and *h*.*B* = (√3/4)⋅4^{2} = 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