Title: READ: Surface Area of Cylinders
Course: Geometry (B)

Cylinders

Learning Objectives

- Find the surface area of cylinders.
- Find the volume of cylinders.
- Find the volume of composite three-dimensional figures.

Introduction

A cylinder is a three-dimensional figure with a pair of parallel and congruent circular ends, or bases. A cylinder has a single curved side that forms a rectangle when laid out flat.

As with prisms, cylinders can be right or oblique. The side of a right cylinder is perpendicular to its circular bases. The side of an oblique cylinder is not perpendicular to its bases.

Surface Area of a Cylinder Using Nets

You can deconstruct a cylinder into a net.

The area of each base is given by the area of a circle:

The area of the rectangular lateral area is given by the product of a width and height. The height is given as . You can see that the width of the area is equal to the circumference of the circular base.

To find the width, imagine taking a can-like cylinder apart with a scissors. When you cut the lateral area, you see that it is equal to the circumference of the can’s top. The circumference of a circle is given by

the lateral area, , is

Now we can find the area of the entire cylinder using .

You can see that the formula we used to find the total surface area can be used for any right cylinder.

Area of a Right Cylinder

The surface area of a right cylinder, with radius and height is given by , where is the area of each base of the cylinder and is the lateral area of the cylinder.

Example 1

Use a net to find the surface area of the cylinder.

First draw and label a net for the figure.

Calculate the area of each base.

Calculate .

Find the area of the entire cylinder.

Thus, the total surface area is approximately

Surface Area of a Cylinder Using a Formula

You have seen how to use nets to find the total surface area of a cylinder. The postulate can be broken down to create a general formula for all right cylinders.

Notice that the base, , of any cylinder is:

The lateral area, , for any cylinder is:

Putting the two equations together we get:

Factoring out a from the equation gives:

The Surface Area of a Right Cylinder

A right cylinder with radius and height can be expressed as:

or:

You can use the formulas to find the area of any right cylinder.

Example 2

Use the formula to find the surface area of the cylinder.

Write the formula and substitute in the values and solve.

Example 3

Find the surface area of the cylinder.

Write the formula and substitute in the values and solve.

Example 4

Find the height of a cylinder that has radius and surface area of .

Write the formula with the given information and solve for .