# Surface Area of a Right Cylinder

How to find the surface area of a right cylinder: definition, formula, proof, example, and its solution.

## Cylinder

This shape is called a cylinder.

## Right Cylinder

A right cylinder is a cylinder

whose centers of the bases

are the endpoints of the height.

## Formula

*A* = 2⋅*πr*^{2} + 2*πr*⋅*h**A*: surface area of a right cylinder*r*: radius of the base*h*: height of the right cylinder

2⋅*πr*^{2}: sum of the base areas

2*πr*⋅*h*: lateral area

## Proof

Draw the net of a right cylinder.

There are two bases.

So the sum of the base areas is 2*B* = 2⋅*πr*^{2}.

Area of a circle

See the lateral face in the net.

The lateral face is a rectangle.

Its width is the circumference (perimeter)

of the base circle:*P* = 2*πr*.

And its height is *h*.

So the lateral area is *Ph* = 2*πr*⋅*h*.

Area of a rectangle

So *A* = 2*B* + *Ph*

= 2⋅*πr*^{2} + 2*πr*⋅*h*.

Surface area of a right prism

## Example

*r* = 4, *h* = 9*A* = 2⋅*π*⋅4^{2} + 2*π*⋅4⋅9