Surface Area of a Cylinder
A closed cylinder has two circular ends and a curved surface that unrolls into a rectangle, so its surface area is \(A = 2\pi r^2 + 2\pi rh\). This NSW Year 9 Advanced maths subtopic uses this formula for closed, open and part cylinders (NESA outcome MA5‑ARE‑C‑01).
You will find exact and rounded surface areas, adjust for cans open at one or both ends, and find the curved area of a pipe, a label and a half-cylinder. You will cost painting a drum and the inside of a tank, find a height from a known area, find the length of a garden roller, and compare the two tubes that can be rolled from one card. It is part of Year 9 Advanced Surface Area and Volume.
Every question with a fully worked solution.
- Surface Area of a Cylinder - Video - Surface Area Of A Cylinder Watch
Theory
A cylinder has two circular ends and a curved surface that unrolls into a rectangle \(2\pi r\) long and \(h\) high. So a closed cylinder has \(\text{SA} = 2\pi r^2 + 2\pi rh\). This Year 9 Advanced maths page covers open and closed cylinders, pipes, half and hollow cylinders, exact answers and working backwards.
A cylinder is a solid with two identical, parallel circular ends joined by a curved surface. Its radius \(r\) is the radius of an end; its height \(h\) (or length) is the distance between the ends.
Cut the curved surface along a straight line and unroll it: it becomes a rectangle. One side is the height; the other wraps once around the circle, so it is the circumference \(2\pi r\). The curved surface area is therefore \(2\pi rh\).
The total surface area adds the circular ends. A closed can has two, a container with no lid has one, and a pipe open at both ends has none.
Answers may be exact (in terms of \(\pi\), such as \(40\pi~\text{cm}^2\)) or rounded. Working in terms of \(\pi\) often makes backwards problems easier, because \(\pi\) cancels.
How to find the surface area of a cylinder
- Find the radius: halve the diameter, and put every length in the same unit.
- Count the ends: closed (two), no lid (one), pipe or label (none).
- Calculate the curved surface \(2\pi rh\) and the ends \(\pi r^2\) each.
- Add, keeping \(\pi\) for an exact answer or rounding at the end.
- Adjust for the context: double for inside and outside, multiply by \(1.2\) for \(20\%\) extra, multiply by a price, or divide by a coverage rate and round up for tins of paint.
Working backwards
- Height from the curved area: \(h = \dfrac{A}{2\pi r}\).
- Height from the total area: subtract the ends, then divide by \(2\pi r\). In exact form, divide every term by \(\pi\) first.
- Radius: if the height is given in terms of \(r\) (such as \(h = 2r\)), substitute it, simplify to a multiple of \(r^2\), and solve.
- Comparing cylinders: write each surface area in terms of \(\pi\) and compare, or set them equal and solve.
(i) Find its surface area exactly and correct to \(2\) decimal places.
(ii) Find the surface area if the tank has no lid.
(i) Halve the diameter: \(r = 1.5\) m. A closed cylinder has two circular ends and a curved surface.
| \(\text{SA}\) | \(=\) | \(2\pi r^2 + 2\pi rh\) |
| \(\) | \(=\) | \(2\pi(1.5)^2 + 2\pi(1.5)(2.2)\) |
| \(\) | \(=\) | \(4.5\pi + 6.6\pi\) |
| \(\) | \(=\) | \(\boldsymbol{11.1\pi} \approx \mathbf{34.87}~\text{m}^2\) |
(ii) Remove one circle, \(\pi r^2 = 2.25\pi\).
| \(\text{SA}\) | \(=\) | \(11.1\pi - 2.25\pi = 8.85\pi \approx \mathbf{27.80}~\text{m}^2\) |
The roof is half the curved surface; the two semicircular ends make one full circle.
| \(\text{roof}\) | \(=\) | \(\dfrac{1}{2} \times 2\pi \times 2 \times 10 = 20\pi\) |
| \(\text{ends}\) | \(=\) | \(2 \times \dfrac{1}{2}\pi \times 2^2 = 4\pi\) |
| \(\text{total}\) | \(=\) | \(24\pi = 75.398\ldots~\text{m}^2\) |
Adding \(15\%\) means multiplying by \(1.15\).
| \(1.15 \times 75.398\ldots\) | \(=\) | \(86.708\ldots \approx \mathbf{86.71}~\text{m}^2\) |
Substitute everything known into the formula. Every term contains \(\pi\), so divide it out.
| \(2\pi(6)^2 + 2\pi(6)h\) | \(=\) | \(168\pi\) |
| \(72\pi + 12\pi h\) | \(=\) | \(168\pi\) |
| \(72 + 12h\) | \(=\) | \(168\) |
| \(12h\) | \(=\) | \(96\) |
| \(h\) | \(=\) | \(\mathbf{8}~\text{cm}\) |
Check: \(72\pi + 12\pi \times 8 = 72\pi + 96\pi = 168\pi\) \(\checkmark\). If the area is given as a decimal instead, subtract the ends first and divide by \(2\pi r\) on the calculator.
Outer radius \(R = 0.6\) m; inner radius \(r = 0.6 - 0.1 = 0.5\) m. List the four surfaces.
| \(\text{outside}\) | \(=\) | \(2\pi(0.6)(0.8) = 0.96\pi\) |
| \(\text{inside}\) | \(=\) | \(2\pi(0.5)(0.8) = 0.8\pi\) |
| \(\text{two ends}\) | \(=\) | \(2\pi(0.6^2 - 0.5^2) = 0.22\pi\) |
| \(\text{SA}\) | \(=\) | \(\boldsymbol{1.98\pi} \approx \mathbf{6.22}~\text{m}^2\) |
Common pitfalls
Frequently asked questions
What is the formula for the surface area of a cylinder?
For a closed cylinder, \(\text{SA} = 2\pi r^2 + 2\pi rh\): two circular ends plus the curved surface.
Why is the curved surface area 2 pi r h?
Unrolled, the curved surface is a rectangle. Its height is \(h\) and its length is the distance around the circle, \(2\pi r\). So its area is \(2\pi rh\).
What is the surface area of an open cylinder?
Leave out the missing end. With no lid it is \(\pi r^2 + 2\pi rh\). A pipe open at both ends has only the curved surface, \(2\pi rh\).
How do you find the height of a cylinder from its surface area?
Substitute the surface area and radius into the formula, subtract the area of the ends, then divide by \(2\pi r\).
What does exact surface area mean?
Leave the answer in terms of \(\pi\), such as \(90\pi~\text{cm}^2\), instead of converting to a decimal.
How do you find the surface area of half a cylinder?
Take half of the curved surface, add the two semicircular ends (together one full circle), and if the flat face is included add the rectangle, diameter \(\times\) length.