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.

A cylinder, outlined in black, is oriented vertically with its top circular face fully visible and its bottom circular face indicated by a dashed outline. The height is labeled $4$4 $m$ . The radius of the bottom circular face is labeled $3$3 $m$ .

Below the cylinder is its net represented by a horizontal rectangle and two circles. One circle is on top of the rectangle and the other circle is below the rectangle. The radius of the circle on top is labeled $3$3 $m$ . The width of the rectangle is labeled $4$4 $m$ .

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

Question 2

Consider the following cylinder.

A cylinder, outlined in green, is oriented horizontally with its right circular face fully visible and its left circular face indicated by a dashed outline. The height is labeled $8$8 $m$ and is marked with a horizontal double-headed arrow across the bottom side. The radius of the circular face is labeled $4$4 $m$ and is marked with a vertical double-headed arrow along the right circular face.

Find the curved surface area of the cylinder to two decimal places.
Using the result from part (a) or otherwise, find the total surface area of the cylinder.

Round your answer to two decimal places.

Question 3

Consider the cylinder shown in the diagram below.

A three-dimensional cylinder with a vertical dimension line to the right of the cylinder indicates its height and is labeled $5$5 cm. A horizontal dimension line below the cylinder indicates the diameter of its circular base and is labeled $6$6 cm. The non-visible part of the cylinder is represented by a dashed line.

Find the surface area of the cylinder in square centimetres.

Round your answer to one decimal place.
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 $m$ ². The total surface area of the cylinder is $25662\pi$25662π $m$ ²

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

Enter each line of working as an equation.
Find the height $h$h of the cylinder.

9.07 Surface area of cylinders