The circumference of a circle is C = πd = 2πr and its area is A = πr². A sector is a fraction θ/360 of the circle, so its arc length is (θ/360) × 2πr and its area is (θ/360) × πr². For a semicircle, the perimeter is the curved arc plus the straight diameter. To find a radius from an area, divide by π and take the square root.
Circle formulae
- Circumference: C = πd = 2πr
- Area: A = πr²
- Arc length = (θ/360) × 2πr
- Sector area = (θ/360) × πr²
- Radius from area: r = √(A ÷ π)
Circumference and area
The radius r goes from the centre to the edge, and the diameter d = 2r goes all the way across. The circumference is the distance around the circle: C = πd or 2πr. The area is A = πr². For a circle of radius 4.5 cm, C = 9π = 28.27 cm and A = 20.25π = 63.62 cm².
Arcs and sectors
A sector is a slice of a circle between two radii, and its curved edge is an arc. A sector with angle θ is θ/360 of the whole circle, so multiply the circumference or area by that fraction. A 72° sector of a circle of radius 10 cm is one fifth of the circle: arc length 4π = 12.57 cm and area 20π = 62.83 cm².
Semicircles and perimeters
A semicircle is half a circle. Its area is ½πr², and its perimeter is half the circumference plus the diameter. Forgetting the straight edge is a very common mistake. For a semicircle of diameter 12 cm, the perimeter is 6π + 12 = 30.85 cm.
Working backwards
If you know the circumference, divide by π to get the diameter. If you know the area, divide by π and then take the square root to find the radius. An area of 200 cm² gives r = √(200 ÷ π) = 7.98 cm.
Exact answers in terms of π
Some questions ask for an answer “in terms of π”. Leave π in the answer, such as 20π cm², rather than using the π button. This is exact, and is often required on non-calculator papers.
Worked examples
Example 1. A circle has radius 4.5 cm. Work out its circumference and area, to 2 decimal places.
- C = 2π × 4.5 = 9π = 28.274…
- A = π × 4.5² = 20.25π = 63.617…
Answer: C = 28.27 cm; A = 63.62 cm².
Example 2. A sector has radius 10 cm and angle 72°. Work out its arc length and area.
- 72/360 = 1/5 of the circle.
- Arc = 1/5 × 20π = 4π. Area = 1/5 × 100π = 20π.
Answer: Arc 12.57 cm; area 62.83 cm² (2 d.p.), or 4π cm and 20π cm².
Example 3. A semicircle has diameter 12 cm. Work out its perimeter, to 2 decimal places.
- Curved part: half of 12π = 6π = 18.85 cm.
- Add the diameter: 18.85 + 12.
Answer: 30.85 cm.
Example 4. A circle has an area of 200 cm². Work out its radius, to 2 decimal places.
- r² = 200 ÷ π = 63.66…
- r = √63.66…
Answer: 7.98 cm.
Example 5. Work out the area of a circle with diameter 14 cm, giving your answer in terms of π.
- r = 7.
- A = π × 7².
Answer: 49π cm².
Common mistakes
- Using the diameter in πr². Halve the diameter to get the radius first.
- Mixing up circumference and area. Circumference is a length (cm); area is squared units (cm²).
- Missing the diameter in a semicircle's perimeter. Add the straight edge to the curved arc.
- Using θ/180 for a sector. A sector is θ/360 of the circle.
- Rounding π early. Use the π button and round only at the end.
How circles are examined
Circle area, circumference, arcs and sectors appear on both tiers of the Edexcel GCSE (1MA1) and the International GCSE (4MA1); sectors and exact answers in terms of π are more common at Higher. Questions ask you to:
- work out the circumference or area of a circle
- find an arc length or sector area
- find the perimeter or area of a semicircle or composite shape
- find a radius or diameter from a given area or circumference
Practise with the circles and circle theorems worksheet, which includes answers, and the geometry revision engine.
Circle formulae are used in cylinders and loci. At Higher tier, see circle theorems.
Frequently asked questions
How do you calculate the area of a circle?
A = πr², where r is the radius.
How do you find the arc length of a sector?
Multiply the circumference by θ/360: arc length = (θ/360) × 2πr.
How do you find the radius from the area?
Divide the area by π, then take the square root.