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Stage 5.1-3

7.03 Surface area of cylinders

Lesson

Surface area of a cylinder

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?

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, where r is the radius.

The image shows a cylinder and its net with with dimensions. Ask your teacher for more information.

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

Exploration

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

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The length of the rectangle that wraps around the cylinder is equal to the circumference of the face of the cylinder.

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}, so the surface area of a cylinder is: \text{Surface area of a cylinder}=2\pi r^{2}+2\pi rh, where r is the radius and h is the height of the cylinder.

Examples

Example 1

Consider the following cylinder.

A cylinder and its net. The radius is 4 metres and the height is 7 metres. Ask your teacher for more information.

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

Worked Solution
Create a strategy

The curved surface area is given by: 2\pi r h.

Apply the idea
\displaystyle \text{Curved surface area}\displaystyle =\displaystyle 2\pi r hWrite the formula
\displaystyle =\displaystyle 2\pi \times 4 \times 7Substitute r and h
\displaystyle \approx\displaystyle 175.93\text{ m}^{2}Evaluate and round

Example 2

Consider the cylinder shown in the diagram below.

A cylinder with a diameter of 6 centimetres and height of 5 centimetres.
a

Find the surface area of the cylinder in square centimetres. Round your answer to one decimal place.

Worked Solution
Create a strategy

Find the radius then use the formula for the surface area of a cylinder.

Apply the idea
\displaystyle r\displaystyle =\displaystyle \dfrac{6}{2}Halve the diameter
\displaystyle =\displaystyle 3\text{ cm}Evaluate
\displaystyle \text{Surface area}\displaystyle =\displaystyle 2\pi r^{2} + 2\pi rhUse the surface area formula
\displaystyle =\displaystyle 2\pi \times 3^{2} + 2\pi \times 3 \times 5Substitute r and h
\displaystyle =\displaystyle 150.8\text{ cm}^{2}Evaluate and round
b

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

Worked Solution
Create a strategy

Use the fact that 1\text{ cm}^{2} = 100\text{ mm}^{2}.

Apply the idea
\displaystyle \text{Surface area}\displaystyle =\displaystyle 150.8 \times 100\text{ mm}^{2}Convert \text{cm}^2 to \text{m}^2
\displaystyle =\displaystyle 15\,080\text{ mm}^{2}Evaluate

Example 3

The area of the circular face on a cylinder is 8281 \pi\text{ m}^2. The total surface area of the cylinder is 25\,662 \pi\text{ m}^2.

a

If the radius of the cylinder is r\text{ m}, find the value of r.

Worked Solution
Create a strategy

We can use the formula: A=\pi r^{2}, and equate the area in the formula with the area given.

Apply the idea
\displaystyle \pi r^{2}\displaystyle =\displaystyle 8281\piSubstitute A
\displaystyle r^{2}\displaystyle =\displaystyle 8281Divide both sides by \pi
\displaystyle r\displaystyle =\displaystyle \sqrt{8281}Take the square root of both sides
\displaystyle r\displaystyle =\displaystyle 91\text{ m}Evaluate
b

Find the height h of the cylinder.

Worked Solution
Create a strategy

We can use the formula: \text{Surface area}=2\pi r^{2}+2\pi rh.

Apply the idea
\displaystyle 25\,662 \pi\displaystyle =\displaystyle 2\pi \times 91^{2} +2\pi h \times 91Substitute r and the surface area
\displaystyle 25\,662 \pi\displaystyle =\displaystyle 16\,562 \pi + 182 \pi hSimplify each term
\displaystyle 182 \pi h\displaystyle =\displaystyle 25\,662 \pi -16\,562 \piSubtract 16\,562 \pi from both sides
\displaystyle 182 \pi h\displaystyle =\displaystyle 9100\piEvaluate the subtraction
\displaystyle \dfrac{182\pi h}{182\pi}\displaystyle =\displaystyle \dfrac{9100\pi}{182\pi}Divide both sides by 182\pi
\displaystyle h\displaystyle =\displaystyle 50\text{ m}Evaluate
Idea summary
\displaystyle \text{Surface area of a cylinder}=2\pi r^{2}+2\pi rh
\bm{r}
is the radius
\bm{h}
is the height of the cylinder

Outcomes

MA5.2-11MG

calculates the surface areas of right prisms, cylinders and related composite solids

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