Circumference of a Circle
Theory
The circumference is the perimeter of a circle: \(C = \pi d\), which is the same as \(C = 2\pi r\). A circle with diameter \(4\) cm has \(C = 4\pi\) cm exactly, about \(12.57\) cm. This Year 8 maths page shows how.
For every circle, circumference \(\div\) diameter \(= \pi\), so \(C = \pi d\). Since \(d = 2r\), also \(C = 2\pi r\). Use the form that matches what you are given. \(\pi \approx 3.14159\).
Exact or rounded? An exact answer leaves \(\pi\) in it: a radius of \(11\) cm gives \(C = 22\pi\) cm. A rounded answer uses the \(\pi\) key and is rounded once, at the end.
Working backwards: \(d = C \div \pi\) and \(r = C \div 2\pi\).
Semicircle: \(P = \dfrac{\pi d}{2} + d\), the curve plus the straight edge. Quadrant: a quarter of the circumference plus two radii.
Composite shapes: add only the edges on the outside; two semicircles on the same diameter make one whole circle.
Rolling: one turn of a wheel moves it one circumference, so turns \(=\) distance \(\div C\), in the same unit.
Circumference:
How to find a circumference
- Choose \(C = \pi d\) or \(C = 2\pi r\) from what you are given.
- Substitute: leave \(\pi\) in for an exact answer, or use the \(\pi\) key.
- Round once, at the end, and give a unit of length.
| \(C\) | \(=\) | \(\pi \times 4 = 4\pi\) |
\(4\pi\) cm
| \(C\) | \(=\) | \(2 \times \pi \times 2\) |
| \(\) | \(=\) | \(12.566\ldots \approx 12.57\) |
\(12.57\) cm
(i) in terms of \(\pi\)
(ii) correct to \(2\) decimal places
| \(C\) | \(=\) | \(2 \times \pi \times 9 = 18\pi\) |
| \(\) | \(=\) | \(56.548\ldots \approx 56.55\) |
(i) \(18\pi\) cm
(ii) \(56.55\) cm
| \(C\) | \(=\) | \(\pi \times 1.4 = 4.398\ldots\) |
\(4.40\) m. Check with the radius: \(2 \times \pi \times 0.7\) gives the same.
(i) its diameter
(ii) its radius
each correct to \(1\) decimal place.
| \(d\) | \(=\) | \(47 \div \pi = 14.96\ldots\) |
| \(r\) | \(=\) | \(47 \div 2\pi = 7.48\ldots\) |
(i) \(15.0\) m
(ii) \(7.5\) m
The \(3\) m edge where the semicircle joins is inside, so it is not counted.
| \(\text{curve}\) | \(=\) | \(\dfrac{1}{2} \times \pi \times 3 = 4.712\ldots\) |
| \(P\) | \(=\) | \(5 + 5 + 3 + 4.712\ldots\) |
| \(\) | \(\approx\) | \(17.71\) |
\(17.71\) m
| \(C\) | \(=\) | \(60\pi \approx 188.5\text{ cm}\) |
| \(150\,000 \div 60\pi\) | \(\approx\) | \(795.8\) |
\(1.5\) km \(= 150\,000\) cm, so \(795\) complete turns.
Common pitfalls
Frequently asked questions
What is the formula for circumference?
\(C = \pi d\) or \(C = 2\pi r\).
What does in terms of pi mean?
Leave \(\pi\) in the answer, such as \(18\pi\) cm.
How do I find the diameter from the circumference?
Divide by \(\pi\): \(d = C \div \pi\).
What is the perimeter of a semicircle?
Half the circumference plus the diameter: \(\dfrac{\pi d}{2} + d\).
Why is the circumference a bit more than 3 diameters?
Because \(\pi\) is about \(3.14\), a little more than \(3\).