Surface area of a cylinder
The surface of a cylinder is made up of circular faces on the top and bottom and a rectangular face that wraps around the curved surface of the cylinder, as shown in the diagram below.
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| A cylinder and its corresponding net | |
One of the dimensions on this rectangle will be the height of the cylinder. The other dimension of the rectangle will be the circumference of the circular base, which can be found using the formula $2\pi r$2πr, where $r$r is the radius of the cylinder.
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| Dimensions of the curved face of a cylinder | |
We can explore this further in the applet below. As the circle rolls, notice how its circumference is equal to the length of the rectangle. Drag the point on the circle, or let the animation roll the circle for you.
Type 1 - cylinder with two closed ends
The surface area of a closed cylinder is the sum of the area of the two circular end faces and the area of the rectangle formed by unrolling the curved face.
$SA=2\times\pi r^2+2\pi r\times h$SA=2×πr2+2πr×h
Type 2 - cylinder with one closed end (like a circular swimming pool)
$SA=\pi r^2+2\pi r\times h$SA=πr2+2πr×h
Type 3 - cylinder with two open ends (like a tunnel)
$SA=2\pi r\times h$SA=2πr×h
If we are given the diameter of the cylinder, d then we first need to calculate the radius, $r=\frac{d}{2}$r=d2, to use the surface area formula.
Worked example
Find the surface area of the cylinder below.
Think: The cylinder will unravel into a rectangle with the dimensions $2\pi r$2πr and $h$h and two circles with radius of $r$r. We can use the formula $SA=2\times\pi r^2+2\pi r\times h$SA=2×πr2+2πr×h to determine the surface area.
Do: The radius of the cylinder is $3$3 m2 and the height is $7$7 m2. Let's substitute these values into the formula:
| $SA$SA | $=$= | $2\times\pi r^2+2\pi r\times h$2×πr2+2πr×h |
The formula for the surface area of a cylinder |
| $=$= | $2\times\pi\times3^2+2\pi\times3\times7$2×π×32+2π×3×7 |
Substitute the values into the formula |
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| $=$= | $60\pi$60π |
Evaluate the multiplication |
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| $=$= | $188.50$188.50 m2 |
Find the approximate answer, rounding to two decimal places |
Practice questions
Question 1
Consider the following cylinder.
Find the area of the curved face of the cylinder.
Round your answer to two decimal places.
Find the area of one of the circular faces.
Round your answer to two decimal places.
Hence find the total surface area of the cylinder.
Round your answer to two decimal places.
Question 2
A cylindrical can of radius $7$7 cm and height $10$10 cm is open at one end. What is the external surface area of the can correct to two decimal places?
Question 3
Ivan is using a toilet paper roll for crafts. He has measured the toilet paper roll to have a diameter $8$8 cm and a length $11$11 cm.
What is the surface area of the toilet paper roll?
Give your answer to the nearest two decimal places.
Surface area of a sphere
A sphere is a solid, three-dimensional object that appears to be circular from any direction. Its surface is defined as the collection of points that are all equal distance from the centre of the sphere.
Unlike the solid objects that we have seen so previously, we cannot unwrap a sphere to get create a net and calculate its area. So the surface area of a sphere is calculated using the formula below:
$\text{SA }=4\pi r^2$SA =4πr2
If we are given the diameter of the cylinder, $d$d, then we first calculate the radius as $r=\frac{d}{2}$r=d2.
Practice questions
Question 4
Find the surface area of the sphere shown.
Round your answer to two decimal places.
Question 5
Find the surface area of the sphere shown.
Round your answer to two decimal places.
Question 6
If two identical spherical balls with radii of $1.4$1.4 m fit exactly inside a cylinder, what is the surface area of the closed cylinder?
Round your answer to one decimal place.
