Surface Area of a Right Cylinder

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.

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.

A right cylinder is a cylinder
whose centers of the bases
are the endpoints of the height.

Formula

A = 2*pi*r^2 + 2*pi*r*h. A: surface area of a right cylinder, r: radius of the base, h: height of the right cylinder

A = 2⋅πr2 + 2πrh

A: surface area of a right cylinder
r: radius of the base
h: height of the right cylinder

2⋅πr2: sum of the base areas
2πrh: lateral area

Proof

Surface Area of a Right Cylinder: Proof of the Formula

Draw the net of a right cylinder.

There are two bases.
So the sum of the base areas is 2B = 2⋅πr2.

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 = 2πr.
And its height is h.

So the lateral area is Ph = 2πrh.

Area of a rectangle

So A = 2B + Ph
= 2⋅πr2 + 2πrh.

Surface area of a right prism

Example

Find the surface area of the given right cylinder. r = 4, h = 9.

r = 4, h = 9

A = 2⋅π⋅42 + 2π⋅4⋅9