The circumference is the distance around the outside of a circle — it is the circle's perimeter. This topic appears on almost every GCSE paper, whether as a standalone question or as part of a problem involving arcs, sectors, or compound shapes.
What Is the Circumference?
The circumference is the total length of the curved boundary of a circle. It is directly proportional to the diameter: no matter how large or small a circle is, the ratio of circumference to diameter is always π (approximately 3.14159).
This gives two equivalent formulas: C = πd (using the diameter) and C = 2πr (using the radius). Since the diameter is twice the radius, these are the same formula written in different ways.
You may be asked to leave your answer in terms of π (an exact answer) or to give a decimal rounded to a specified number of decimal places or significant figures.
Key Formulas
Step-by-Step Method
- Identify whether you are given the radius or the diameter.
- If given the radius, either double it to get the diameter and use C = πd, or use C = 2πr directly.
- Substitute into the formula and calculate — leave in terms of π or round as instructed.
Worked Example 1 — Foundation Level
Question: A circle has a diameter of 14 cm. Find the circumference. Give your answer to 1 decimal place.
Working: C = πd C = π × 14 C = 43.982… C = 44.0 (1 d.p.)
Answer: 44.0 cm
Worked Example 2 — Higher Level
Question: A circle has a circumference of 75 cm. Find the radius. Give your answer to 2 decimal places.
Working: C = 2πr 75 = 2πr r = 75 ÷ (2π) r = 75 ÷ 6.2832… r = 11.94 (2 d.p.)
Answer: 11.94 cm
Worked Example 3 — Exam Style
Question: A semicircular window has a diameter of 60 cm. Calculate the perimeter of the window.
Working: Curved part = half the circumference = ½ × πd = ½ × π × 60 = 30π Straight edge = diameter = 60 Perimeter = 30π + 60 = 94.248… + 60 = 154.2 (1 d.p.)
Answer: 154.2 cm
Common Mistakes
- Confusing radius and diameter. The diameter is twice the radius. Using the wrong one will double or halve your answer. Always check which measurement the question gives.
- Forgetting the straight edge in semicircles. The perimeter of a semicircle includes the curved part plus the diameter — not just the half circumference.
- Rounding π too early. Use the π button on your calculator rather than 3.14, which introduces rounding errors.
- Omitting units. Circumference is a length, so the unit is cm, m, mm, etc. — not cm².
Exam Tips
- If the question says "give your answer in terms of π," leave π in your answer (e.g., 14π cm) without calculating a decimal.
- On non-calculator papers, use π ≈ 3.14 or the value given in the question.
- Double-check whether you have been given the radius or the diameter before substituting.
Practice Questions
Q1 (Foundation): A circle has a radius of 5 cm. Find its circumference in terms of π.
Q2 (Foundation): A circle has a diameter of 20 cm. Find its circumference to 1 decimal place.
Q3 (Higher): The circumference of a circle is 50π cm. Find the diameter and the radius.
Practise circumference of a circle questions with instant feedback — completely free on GCSEMathsAI.
Related Topics
Summary
- The circumference is the perimeter of a circle.
- Use C = πd or equivalently C = 2πr.
- To find the radius from the circumference, rearrange to r = C ÷ (2π).
- Always check whether you have the radius or the diameter before substituting.
- For semicircles and quarter-circles, remember to add the straight edges to the curved part.
- Leave answers in terms of π when the question requests an exact answer.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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