Circles: area, circumference, arcs and sectors
Two formulas cover circles: one for the distance round the edge and one for the space inside. Arcs and sectors are fractions of those, using the angle at the centre out of 360°.
Part 1 of 3: Learn it
In short
- Circumference C = πd = 2πr.
- Area A = πr². Square the radius, not the diameter.
- Arc and sector: multiply the full-circle answer by θ/360.
Where this is in your specification
Spec points: DfE G17 and N8 (arcs and sectors, G18, are Higher tier)
| Board | Topic: Area, perimeter and volume |
|---|---|
| DfE content | G12, G13, G14, G16, G17, G18, N8 |
| AQA 8300 | Geometry and measures |
| Edexcel 1MA1 | Geometry and measures |
| OCR J560 | 10 Mensuration |
| Eduqas C300 | HG12, HG14, HG16 to HG18 |
| Cambridge IGCSE 0580 | E5.1 to E5.5 |
| National 5 C847 75 | Geometric skills: volumes of solids, arcs and sectors |
The formulas
The radius is half the diameter. Check which one the question gives you before you start.
"Give your answer in terms of π" means leave π in the answer: a circle of radius 5 has area 25π cm².
Semicircles and quarter circles
Halve or quarter the area. For the perimeter, take the curved part (half or a quarter of the circumference) and add the straight edges: the diameter for a semicircle, two radii for a quarter circle.
Arcs and sectors (Higher)
The perimeter of a sector is the arc length plus two radii.
Find the circumference of a circle with diameter 9 cm, to 1 decimal place.
Show the answer
9π = 28.3 cm.
Part 2 of 3: See it worked
Worked examples
Example 1
A circle has radius 6 cm. Find its circumference and area to 1 decimal place.
- C = 2 × π × 6 = 12π = 37.699...
- A = π × 6² = 36π = 113.097...
Answer: Circumference 37.7 cm, area 113.1 cm².
Example 2
A sector has radius 10 cm and angle 72°. Find its area and perimeter to 1 decimal place.
- 72/360 = 1/5
- Area = 1/5 × π × 100 = 20π = 62.83 cm²
- Arc = 1/5 × 2π × 10 = 4π = 12.57 cm
- Perimeter = 12.57 + 10 + 10 = 32.57 cm
Answer: Area 62.8 cm², perimeter 32.6 cm.
Example 3
Find the perimeter of a semicircle with diameter 14 cm.
- Curved part: ½ × π × 14 = 7π = 21.99 cm
- Add the diameter: 21.99 + 14 = 35.99
Answer: 36.0 cm to 1 decimal place.
Common mistakes
- Using the diameter in A = πr².
- Working out (πr)² instead of π × r².
- Leaving out the straight edges when finding the perimeter of a semicircle or sector.
- Rounding part way through. Keep the full value on the calculator until the end.
A circle has circumference 10π cm. Find its area in terms of π.
Show the answer
d = 10, so r = 5 and the area is 25π cm².
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Find the circumference of a circle with diameter 9 cm, to 1 decimal place.
9π = 28.3 cm.
A circle has circumference 10π cm. Find its area in terms of π.
d = 10, so r = 5 and the area is 25π cm².
Find the area of a quarter circle of radius 8 cm in terms of π.
¼ × π × 64 = 16π cm².
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses