Circles appear everywhere in maths and the real world, from bicycle wheels to pizza bases. To describe a circle mathematically you need two key measurements: its circumference (the distance around the edge) and its area (the space inside). Both depend on a single remarkable number: π (pi), approximately 3.14159.
What is the circumference of a circle?
The circumference is the perimeter of a circle — the total distance around its edge.
There are two equivalent formulae, depending on whether you know the radius or the diameter:
- C = 2πr (when you know the radius, r)
- C = πd (when you know the diameter, d)
These are identical because the diameter is exactly twice the radius: d = 2r.
Worked Example 1 — Find the circumference when r = 5 cm.
- Write the formula: C = 2πr
- Substitute: C = 2 × π × 5
- Simplify: C = 10π
- Calculate: C ≈ 10 × 3.14159 ≈ 31.4 cm (to 3 significant figures)
If the question asks you to leave your answer in terms of π, stop at step 3 and write C = 10π cm.
Worked Example 2 — Find the circumference when d = 12 cm.
- Write the formula: C = πd
- Substitute: C = π × 12
- Simplify: C = 12π
- Calculate: C ≈ 12 × 3.14159 ≈ 37.7 cm (to 3 significant figures)
What is the area of a circle?
The area of a circle is the amount of space enclosed inside it.
The formula is:
A = πr²
This is read as "pi r squared". Notice that you always need the radius, and you must square it before multiplying by π.
Worked Example 3 — Find the area when r = 4 cm.
- Write the formula: A = πr²
- Substitute: A = π × 4²
- Square the radius: A = π × 16
- Simplify: A = 16π
- Calculate: A ≈ 16 × 3.14159 ≈ 50.3 cm² (to 3 significant figures)
Always include the correct units squared (cm², m²) for an area answer.
How do I find the radius from the area?
Sometimes you are given the area and asked to find the radius. Rearrange A = πr² step by step:
- Start with: A = πr²
- Divide both sides by π: A ÷ π = r²
- Square root both sides: r = √(A ÷ π)
Worked Example 4 — Find the radius when A = 100 cm².
- r² = 100 ÷ π
- r² = 100 ÷ 3.14159 ≈ 31.831
- r = √31.831 ≈ 5.64 cm (to 3 significant figures)
How do I find the diameter from the circumference?
Rearrange C = πd to make d the subject:
- Start with: C = πd
- Divide both sides by π: d = C ÷ π
Worked Example 5 — Find the diameter when C = 45 cm.
- d = 45 ÷ π
- d ≈ 45 ÷ 3.14159 ≈ 14.3 cm (to 3 significant figures)
What is the difference between diameter and radius?
This distinction trips up many students, so it is worth being crystal clear:
| Term | Definition | Symbol | Relationship |
|---|---|---|---|
| Radius | Distance from the centre to the edge | r | Half of the diameter |
| Diameter | Distance straight across the circle through the centre | d | Twice the radius: d = 2r |
| Circumference | Distance around the edge | C | C = 2πr = πd |
| Area | Space enclosed inside the circle | A | A = πr² |
A quick memory check: if a question gives you the diameter, always halve it to get the radius before using A = πr².
What are the most common mistakes with circles?
Mistake 1: Using the diameter instead of the radius in the area formula.
If d = 10 cm, then r = 5 cm. The area is π × 5² = 25π ≈ 78.5 cm². A common error is to write π × 10² = 100π, which is four times too large.
Mistake 2: Forgetting to square the radius.
A = πr² means you square r first, then multiply by π. Writing A = π × r (without squaring) gives a completely wrong answer.
Mistake 3: Giving area units without the square.
Area is always in square units (cm², m², mm²). Writing "cm" instead of "cm²" will lose a mark.
Mistake 4: Using an approximate value for π too early.
Unless told otherwise, use the π button on your calculator for the most accurate answer, and only round at the final step.
Frequently asked questions
What is π and why does it appear in both formulae?
π (pi) is the ratio of a circle's circumference to its diameter, and it is the same for every circle ever drawn: approximately 3.14159. Because both the circumference and area depend on the fundamental shape of a circle, π appears in both formulae. It is an irrational number, meaning its decimal expansion never repeats or terminates.
When should I leave my answer in terms of π?
Leave your answer in terms of π (for example, 16π cm²) whenever the question says "give your answer in terms of π" or "leave your answer in exact form". This avoids any rounding error and is considered the exact answer. If the question asks you to "calculate" or gives a decimal context, use the π button and round as instructed.
Does it matter whether I use 3.14 or the π button?
Yes — always use the π button on your calculator unless the question specifically tells you to use 3.14 or 22/7. Using 3.14 introduces a small rounding error that can cost marks in multi-step problems, particularly at GCSE. The π button stores far more decimal places than any approximation.
How do I know whether a question is about circumference or area?
Think about what is being measured. Circumference is a length (one-dimensional) — you use it for fencing, edging, or the distance around something. Area is a flat surface (two-dimensional) — you use it for painting, covering, or fitting something inside the circle. The units are your clue: circumference is in cm or m; area is in cm² or m².
Ready to test yourself with instant feedback? Practice circle problems with Professor Pi on AI Tutors — your personal KS3 maths tutor.