Surface Area of a Cylinder
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Theory
The surface area of a cylinder is its two circular ends plus the curved surface. Unrolled, the curved surface is a rectangle with area \(2\pi rh\). This Year 9 maths page covers closed cylinders, cylinders open at one or both ends, half and hollow cylinders, exact answers in terms of \(\pi\), and working backwards to the height.
A cylinder has two identical circular ends joined by a curved surface. The radius of the ends is \(r\) and the height is \(h\).
Cut the curved surface down the side and flatten it, like peeling a label off a tin. It becomes a rectangle. Its width is the distance round the end, the circumference \(2\pi r\), and its height is \(h\).
A closed cylinder, such as a tin, has both ends. A cup or a tank with no lid is open at one end: one circle. A pipe is open at both ends: no circles, just the curved surface.
An exact answer is left in terms of \(\pi\), such as \(66\pi \text{ cm}^2\).
| Cylinder | Circles | Surface area |
|---|---|---|
| closed (a tin) | \(2\) | \(2\pi r^2 + 2\pi rh\) |
| open at one end (a cup, a tank) | \(1\) | \(\pi r^2 + 2\pi rh\) |
| open at both ends (a pipe, a label) | \(0\) | \(2\pi rh\) |
How to find the surface area of a cylinder
- Find the radius. If you are given the diameter, halve it. Change to the units the answer needs.
- Count the circular ends: closed \(2\), open at one end \(1\), a pipe \(0\).
- Work out the ends (\(\pi r^2\) each) and the curved surface (\(2\pi rh\)).
- Add them. Leave \(\pi\) in for an exact answer; otherwise round once at the end.
Halve the diameter first: \(r = 7\) cm. A closed cylinder has two circular ends and the curved surface.
| \(\text{ends}\) | \(=\) | \(2 \times \pi \times 7^2 = 98\pi\) |
| \(\text{curved}\) | \(=\) | \(2 \times \pi \times 7 \times 9 = 126\pi\) |
| \(SA\) | \(=\) | \(224\pi = 703.716\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{703.72} \text{ cm}^2\) |
No lid, so only one circle: the base.
| \(\text{base}\) | \(=\) | \(\pi \times 2.5^2 = 19.634\ldots\) |
| \(\text{curved}\) | \(=\) | \(2 \times \pi \times 2.5 \times 4 = 62.831\ldots\) |
| \(SA\) | \(=\) | \(82.466\ldots \approx \mathbf{82.47} \text{ m}^2\) |
Adding a second circle would make the answer too big.
(i) Find the exact surface area of a closed cylinder with radius \(3\) cm and height \(8\) cm.
(ii) A pipe, open at both ends, has a diameter of \(40\) cm and a length of \(1.5\) m. Find its curved surface area in square metres, to two decimal places.
(i) "Exact" means leave \(\pi\) in the answer. Write each part as a number of \(\pi\)s, then add.
| \(\text{ends}\) | \(=\) | \(2\pi \times 3^2 = 18\pi\) |
| \(\text{curved}\) | \(=\) | \(2\pi \times 3 \times 8 = 48\pi\) |
| \(SA\) | \(=\) | \(\boldsymbol{66\pi} \text{ cm}^2\) |
(ii) A pipe has no ends. Change to metres first: \(d = 0.4\) m, so \(r = 0.2\) m.
| \(A\) | \(=\) | \(2 \times \pi \times 0.2 \times 1.5\) |
| \(\) | \(=\) | \(0.6\pi = 1.884\ldots \approx \mathbf{1.88} \text{ m}^2\) |
The label is the curved surface, so use \(A = 2\pi rh\). Substitute, then divide.
| \(150.80\) | \(=\) | \(2 \times \pi \times 3 \times h\) |
| \(150.80\) | \(=\) | \(18.849\ldots \times h\) |
| \(h\) | \(=\) | \(\dfrac{150.80}{18.849\ldots}\) |
| \(\) | \(\approx\) | \(\mathbf{8} \text{ cm}\) |
Check: \(2 \times \pi \times 3 \times 8 = 150.796\ldots\), which rounds to \(150.80\). For a closed cylinder, take away the two ends first, then divide by \(2\pi r\).
Common pitfalls
Frequently asked questions
What is the formula for the surface area of a cylinder?
For a closed cylinder, \(SA = 2\pi r^2 + 2\pi rh\): two circular ends plus the curved surface.
What is the curved surface area of a cylinder?
It is \(2\pi rh\). Unrolled, the curved surface is a rectangle as wide as the circumference \(2\pi r\) and as tall as \(h\).
How do you find the surface area of an open cylinder?
Leave out the missing ends. Open at one end: \(\pi r^2 + 2\pi rh\). Open at both ends, like a pipe: just \(2\pi rh\).
What does exact surface area mean?
It means leaving \(\pi\) in the answer instead of rounding. A closed cylinder with \(r = 3\) cm and \(h = 8\) cm has \(SA = 66\pi \text{ cm}^2\).
How do you find the height of a cylinder from its surface area?
Substitute the surface area and radius into the formula and solve for \(h\). For a curved surface only, \(h = \dfrac{A}{2\pi r}\).
How do you find the surface area of half a cylinder?
Take half of the curved surface, \(\pi rh\), add the two semicircular ends (together one circle, \(\pi r^2\)), and if it is closed add the flat rectangle, \(2r \times h\).