7.03 Surface area of cylinders

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.

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.

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.

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 h	Write the formula
	\displaystyle =	\displaystyle 2\pi \times 4 \times 7	Substitute r and h
	\displaystyle \approx	\displaystyle 175.93\text{ m}^{2}	Evaluate and round

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Example 2

Consider the cylinder shown in the diagram below.

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 rh	Use the surface area formula
	\displaystyle =	\displaystyle 2\pi \times 3^{2} + 2\pi \times 3 \times 5	Substitute r and h
	\displaystyle =	\displaystyle 150.8\text{ cm}^{2}	Evaluate and round

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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

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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\pi	Substitute A
\displaystyle r^{2}	\displaystyle =	\displaystyle 8281	Divide 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

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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 91	Substitute r and the surface area
\displaystyle 25\,662 \pi	\displaystyle =	\displaystyle 16\,562 \pi + 182 \pi h	Simplify each term
\displaystyle 182 \pi h	\displaystyle =	\displaystyle 25\,662 \pi -16\,562 \pi	Subtract 16\,562 \pi from both sides
\displaystyle 182 \pi h	\displaystyle =	\displaystyle 9100\pi	Evaluate 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

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