Middle Years

# 9.07 Surface area of cylinders

Lesson

A cylinder has three faces: two identical circular bases and a curved surface that joins the two bases together.

The surface area of a cylinder is the sum of the areas of these three faces. We already know how to find the area of the circular bases, but what about the curved surface?

### The curved surface area of a cylinder

By "unwrapping" the cylinder we can treat the curved surface as a rectangle, with one side length equal to the height of the cylinder, and the other the perimeter (circumference) of the base circle. This is given by $2\pi r$2πr, where $r$r is the radius.

This means the surface area of the curved part of a cylinder is $2\pi rh$2πrh, where $r$r is the radius and $h$h is the height.

We can see how the cylinder unrolls to make this rectangle in the applet below:

To find the surface area of the whole cylinder, we need to add the area of the top and bottom circles to the area of the curved part. Both of these circles have an area of $\pi r^2$πr2, so the surface area of a cylinder is:

Surface area of a cylinder

$\text{Surface area of a cylinder}=2\pi r^2+2\pi rh$Surface area of a cylinder=2πr2+2πrh

Where $r$r is the radius and $h$h is the height of the cylinder.

#### Practice questions

##### Question 1

Consider the following cylinder.

1. Find the curved surface area of the cylinder to two decimal places.

##### Question 2

Consider the following cylinder.

1. Find the curved surface area of the cylinder to two decimal places.

2. Using the result from part (a) or otherwise, find the total surface area of the cylinder.

##### Question 3

Consider the cylinder shown in the diagram below.

1. Find the surface area of the cylinder in square centimetres.

2. Use your answer from part (a) to find the surface area of the cylinder in square millimetres?

##### Question 4

The area of the circular face on a cylinder is $8281\pi$8281π m2. The total surface area of the cylinder is $25662\pi$25662π m2

1. If the radius of the cylinder is $r$r m, find the value of $r$r.

Enter each line of working as an equation.

2. Hence, find the height $h$h of the cylinder.