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).
Ces fiches sont en anglais car elles suivent les programmes d'examen britanniques.
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses