Volume of a Cylinder
A cylinder is a stack of identical circles, so its volume is the area of the circular base times the height: \(V = \pi r^2 h\). This NSW Year 8 maths subtopic (NESA outcome MA4‑VOL‑C‑01) gives answers exactly in terms of \(\pi\) or rounded, and halves a diameter before squaring it.
You will find the volume of cylinders and half cylinders, change the answer to millilitres, litres or kilolitres, and work backwards to a height or radius. You will also see why doubling the diameter makes the volume four times as large, compare a square and a round baking tin, and count how many bottles a water cooler fills. It is part of Year 8 Volume.
Theory
A cylinder is a prism with a circular cross-section, so \(V = Ah\) becomes \(V = \pi r^2 h\). A cylinder with radius \(2\) cm and height \(5\) cm has \(V = \pi \times 4 \times 5 = 20\pi~\text{cm}^3\). This Year 8 maths page shows how.
A cylinder is a stack of identical circles. Its cross-section has area \(\pi r^2\), so \(V = \pi r^2 h\), where \(r\) is the radius of the base and \(h\) the perpendicular height.
A diameter is not a radius: halve it, \(r = d \div 2\).
Exact or rounded. An exact answer keeps \(\pi\), such as \(45\pi~\text{cm}^3\); otherwise multiply by \(\pi\) on the last line. Use \(\dfrac{22}{7}\) only if told to.
Half a cylinder (a trough or gutter): \(V = \dfrac{1}{2}\pi r^2 h\).
Capacity: \(1~\text{cm}^3 = 1\) mL, \(1000~\text{cm}^3 = 1\) L, \(1~\text{m}^3 = 1\) kL.
Working backwards: \(h = V \div (\pi r^2)\), or \(r^2 = V \div (\pi h)\) and then the square root.
Changing a measurement. \(r\) is squared but \(h\) is not: trebling the radius makes the volume \(9\) times as big; trebling the height makes it \(3\) times as big.
Volume of a cylinder:
How to find the volume of a cylinder
- Find the radius (halve a diameter) and the height, in one unit.
- Substitute into \(V = \pi r^2 h\), squaring \(r\) first.
- Give an exact answer, or round at the end; convert to litres if asked.
| \(V\) | \(=\) | \(\pi \times 2^2 \times 5\) |
| \(\) | \(=\) | \(20\pi\) |
\(20\pi~\text{cm}^3\)
| \(50\pi\) | \(=\) | \(\pi r^2 \times 2\) |
| \(r^2\) | \(=\) | \(25\) |
| \(r\) | \(=\) | \(5\) |
\(r = 5\) cm (a length is positive).
(i) in terms of \(\pi\)
(ii) to \(1\) decimal place
| \(V\) | \(=\) | \(\pi \times 36 \times 9 = 324\pi\) |
| \(\) | \(=\) | \(1017.87\ldots\) |
(i) \(324\pi~\text{cm}^3\)
(ii) \(1017.9~\text{cm}^3\)
| \(r\) | \(=\) | \(11\) |
| \(V\) | \(=\) | \(\pi \times 121 \times 30 = 3630\pi\) |
| \(\) | \(=\) | \(11\,403.98\ldots~\text{cm}^3\) |
\(11.40\) L (divide by \(1000\)).
| \(V\) | \(=\) | \(\dfrac{1}{2} \times \pi \times 6^2 \times 300\) |
| \(\) | \(=\) | \(5400\pi = 16\,964.6\ldots\) |
\(16\,965~\text{cm}^3\), about \(17\) L.
| \(10\pi r^2\) | \(=\) | \(500\) |
| \(r^2\) | \(=\) | \(\dfrac{500}{10\pi} = 15.915\ldots\) |
| \(r\) | \(=\) | \(3.989\ldots\) |
\(4.0\) cm
Common pitfalls
Frequently asked questions
What is the formula for the volume of a cylinder?
\(V = \pi r^2 h\).
How do I find the volume from the diameter?
Halve the diameter to get the radius, then use \(V = \pi r^2 h\).
How many litres does a cylinder hold?
Find the volume in cm\(^3\), then divide by \(1000\).
How do I find the height from the volume?
Divide the volume by the base area: \(h = V \div \pi r^2\).
What happens if the height is tripled?
The volume is tripled too, because \(h\) is not squared.