Circumference of a circle: formula and worked examples
The circumference of a circle is the distance all the way round it: C = πd, or C = 2πr. This guide shows how to use the formula, how to work backwards to the radius or diameter, and how to handle semicircles, wheels and compound shapes.
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The circumference formula
C = πd = 2πr, where d is the diameter and r is the radius. Both versions say the same thing, because the diameter is twice the radius.
π (pi) is the number of times the diameter fits round the circumference: about 3.14159 for every circle, whatever its size. Use the π button on your calculator rather than 3.14, unless the question tells you otherwise.
| You know | Circumference | Example |
|---|---|---|
| The radius r | C = 2πr | r = 4 cm gives C = 8π ≈ 25.1 cm |
| The diameter d | C = πd | d = 12 cm gives C = 12π ≈ 37.7 cm |
| The area A | First r = √(A/π), then C = 2πr | A = 49π gives r = 7 and C = 14π |
Worked examples
Example 1: from the radius
Find the circumference of a circle of radius 4 cm, to 1 decimal place.
- C = 2πr = 2 × π × 4 = 8π.
- C ≈ 25.1 cm.
Example 2: from the diameter
A circular pond has a diameter of 12 m. How far is it round the edge?
- C = πd = π × 12 = 12π.
- C ≈ 37.7 m.
Example 3: working backwards
A circle has a circumference of 50 cm. Find its diameter and radius.
- C = πd, so d = C/π = 50/π ≈ 15.9 cm.
- r = d/2 ≈ 7.96 cm. Divide by π, do not multiply.
Example 4: the perimeter of a semicircle
Find the perimeter of a semicircle with diameter 10 cm.
- The curved part is half the circumference: (1/2) × π × 10 = 5π ≈ 15.71 cm.
- Add the straight edge, the diameter: 15.71 + 10 = 25.71 cm. Forgetting the diameter is the most common mistake.
Example 5: a quarter circle
Find the perimeter of a quarter circle with radius 6 cm.
- The arc is a quarter of 2π × 6: 3π ≈ 9.42 cm.
- Add the two radii: 9.42 + 6 + 6 = 21.42 cm.
Example 6: a wheel
A bicycle wheel has a diameter of 70 cm. How many complete turns does it make in 1 km?
- One turn moves the bike one circumference: π × 70 ≈ 219.9 cm.
- 1 km = 100,000 cm, and 100,000 ÷ 219.9 ≈ 454.7.
- So it makes 454 complete turns, and needs part of a 455th to reach 1 km.
Example 7: a running track
A track has two straights of 100 m joined by semicircular ends of diameter 60 m. Find the length of one lap.
- The two semicircles make one full circle: π × 60 ≈ 188.5 m.
- Add the straights: 188.5 + 200 = 388.5 m.
Estimating without a calculator
To check an answer, round π to 3. A circle of radius 5 cm has circumference about 2 × 3 × 5 = 30 cm, so an answer of 31.4 cm is sensible and 78.5 cm (the area) is not. The circumference is always a little more than three times the diameter.
Why the circumference is π times the diameter
Every circle is an enlargement of every other circle, so the ratio of circumference to diameter is the same for all of them. That ratio is π. It cannot be written exactly as a fraction or a terminating decimal, which is why exact answers keep the symbol π. 22/7 is a close fraction, often used for estimates, but it is not exact.
Exact answers: leaving it in terms of π
On a non-calculator paper, or when the question says "give your answer in terms of π", leave π as a symbol. A circle of radius 5 cm has circumference 10π cm. Writing 31.4 instead can lose the final mark. Our guide to calculator and non-calculator topics has more on exact answers.
Circumference, arc length and area
- Arc length is a fraction of the circumference: for an angle θ at the centre, arc = (θ/360) × 2πr. See area of a sector and arc length.
- Area is πr², measured in square units. Circumference is a length, so it has no square. See area of a circle.
- Compound shapes: add only the edges on the outside of the shape, never the lines inside it.
Practice questions
Try each one before opening the answer.
Q1. Find the circumference of a circle with radius 9 cm, to 1 decimal place.
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Q2. A circular table has a diameter of 2.5 m. Find its circumference to 2 decimal places.
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Q3. A circle has a circumference of 31.4 cm. Find its radius to 2 decimal places.
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Q4. Find the perimeter of a semicircle of radius 3 cm, to 2 decimal places.
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Q5. A circle has area 49π cm². Find its circumference in terms of π.
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Common mistakes
- Using the diameter in C = 2πr, which doubles the answer.
- Forgetting to add the straight edge when finding the perimeter of a semicircle or quarter circle.
- Multiplying by π when working backwards from the circumference, instead of dividing.
- Rounding π to 3.14 when the question expects the calculator value.
Frequently asked questions
What is the formula for the circumference of a circle?
C = πd, or C = 2πr, where d is the diameter and r the radius.
How do you find the radius from the circumference?
Divide the circumference by 2π: r = C/(2π). To find the diameter, divide by π.
What is the difference between circumference and perimeter?
Circumference is the perimeter of a circle. For other shapes, and for parts of circles, we say perimeter.
Is the circumference formula given in the GCSE exam?
For GCSE Maths exams in 2025, 2026 and 2027 a formulae sheet is provided. Learn the formula anyway: you will use it in many questions and looking it up costs time.
Key takeaways
- C = πd = 2πr.
- Work backwards by dividing by π (for d) or 2π (for r).
- For semicircles and sectors, add the straight edges.