Number
Measures & Accuracy: Error Intervals, Bounds, Rounding & Unit Conversions (AQA 8300)
Practice for Foundation and Higher tiers, with full worked solutions.
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Circles, arcs & sectors · revision guide
Arc length and sector area
One idea, three formulas and six worked examples: arc length, sector area and perimeter, major sectors, finding the angle and segments, with and without a calculator.
Arc length and sector area come up again and again on Cambridge IGCSE 0580 papers. In the 63 question papers from June 2024 to June 2026 that we checked, at least 17 have an arc or sector question, and they cluster on the Extended papers. Since 2025, Paper 2 has been non-calculator, and 9 of the 10 non-calculator Extended Paper 2s in that period include one. So you need two skills: working accurately with a calculator, and giving exact answers in terms of π without one.
The good news is that it all comes from one idea.
The one idea: a sector is a fraction of a circle
A sector is a slice of a circle between two radii, like a slice of pizza. Its angle at the centre, θ, tells you what fraction of the whole circle it is:
fraction of the circle = θ / 360
So the arc is that fraction of the circumference, and the sector is that fraction of the area:
Arc length = θ/360 × 2πr
Sector area = θ/360 × πr²
Sector perimeter = arc length + r + r
Learn these. The 0580 List of formulas on page 2 of the paper gives the circumference and area of a circle, but not arc length or sector area. The AQA GCSE formula sheets don't include them either.
Words you need
- Arc: part of the circumference. The minor arc is the shorter one, the major arc the longer one.
- Sector: the region between two radii and an arc. A minor sector has an angle under 180°; a major sector has an angle over 180°.
- Chord: a straight line joining two points on the circle.
- Segment: the region between a chord and an arc.
Arc length
Example 1 · Core and Foundation
A sector has radius 10 cm and angle 72°. Find the arc length. Give your answer in terms of π, then to 3 significant figures.
- Fraction of the circle: 72/360 = 1/5.
- Circumference = 2 × π × 10 = 20π.
- Arc = 1/5 × 20π = 4π cm.
- With a calculator: 4π = 12.566… = 12.6 cm (3 s.f.).
Answer: 4π cm, or 12.6 cm
Sector area
Example 2 · Core and Foundation · non-calculator
Find the area of a sector with radius 6 cm and angle 120°. Give your answer in terms of π.
- Fraction: 120/360 = 1/3.
- Area of the whole circle = π × 6² = 36π.
- Sector = 1/3 × 36π = 12π cm².
Answer: 12π cm² (37.7 cm² to 3 s.f.)
Without a calculator, simplify the fraction first. 1/3 of 36π is easy; 120 × 36 ÷ 360 is not. Knowing these by heart saves time:
| Angle | Fraction of the circle |
|---|---|
| 30° | 1/12 |
| 40° | 1/9 |
| 45° | 1/8 |
| 60° | 1/6 |
| 72° | 1/5 |
| 90° | 1/4 |
| 120° | 1/3 |
| 135° | 3/8 |
| 150° | 5/12 |
| 180° | 1/2 |
| 270° | 3/4 |
At IGCSE Core the sector angle is always a factor of 360 (such as 30°, 45°, 60°, 72°, 90° or 120°), so the fraction is always 1 over a whole number.
Perimeter of a sector
The perimeter is the distance all the way round: the curved arc plus the two straight radii. Forgetting the radii is the most common mistake on these questions.
Example 3 · Core and Foundation
Find the perimeter of a sector with radius 9 cm and angle 80°.
- Fraction: 80/360 = 2/9.
- Arc = 2/9 × 2 × π × 9 = 4π cm.
- Perimeter = arc + 9 + 9 = 4π + 18 cm.
- With a calculator: 12.566… + 18 = 30.566… = 30.6 cm.
Answer: (4π + 18) cm, or 30.6 cm
Major sectors
If the question asks for the major sector but the diagram shows the small angle, subtract from 360° first.
Example 4 · Extended and Higher · non-calculator
A circle has radius 4 cm. The minor sector has angle 45°. Find the area of the major sector, in terms of π.
- Major sector angle = 360° − 45° = 315°.
- Fraction: 315/360 = 7/8.
- Area of the circle = π × 4² = 16π.
- Major sector = 7/8 × 16π = 14π cm².
Answer: 14π cm² (44.0 cm² to 3 s.f.)
Check: the minor sector is 1/8 × 16π = 2π, and 14π + 2π = 16π, the whole circle. ✓
Working backwards: finding the angle or the radius
Put the numbers you know into the formula and solve for the unknown.
Example 5 · Extended and Higher · calculator
A sector has radius 5 cm and arc length 7.5 cm. Find the angle of the sector.
- Arc = θ/360 × 2πr, so 7.5 = θ/360 × 2 × π × 5 = θ/360 × 10π.
- Divide both sides by 10π: θ/360 = 7.5 ÷ 10π = 0.2387…
- Multiply by 360: θ = 85.94…°.
Answer: θ = 85.9°
To find a radius from a sector area, rearrange in the same way: r² = area × 360 ÷ (θπ), then square root.
Segments
A segment is the part of a sector that is left when you remove the triangle. So segment area = sector area − triangle area.
Example 6 · Extended and Higher
A sector has radius 6 cm and a right angle at the centre. Find the area of the shaded segment.
- Sector area = 1/4 × π × 6² = 9π.
- The triangle has a right angle, so its area = ½ × 6 × 6 = 18.
- Segment = 9π − 18 = 10.27…
Answer: (9π − 18) cm², or 10.3 cm²
When the angle is not 90°, the triangle's area comes from ½ab sin C, which is Extended and Higher content.
Mistakes that lose marks
- Using the diameter as the radius. If the question gives the diameter, halve it first.
- Mixing up the formulas. Arc length uses 2πr (a length); sector area uses πr² (an area, so cm²).
- Forgetting the two radii in a perimeter. Perimeter = arc + 2r.
- Using the minor angle for a major sector. Use 360° − θ.
- Rounding too early, or using π = 3.14. 0580 papers say “use either your calculator value or 3.142”. Keep the full value until the end, then give 3 significant figures.
- Not leaving π in. “In terms of π” means an answer like 12π, not 37.7.
What is on your syllabus
| Syllabus | What you need |
|---|---|
| IGCSE 0580 Core (C5.3) | Arc length and sector area as fractions of the circumference and area, where the sector angle is a factor of 360°. Answers may be in terms of π. |
| IGCSE 0580 Extended (E5.3) | Any sector angle, including minor and major sectors. Answers may be in terms of π. |
| GCSE AQA 8300 (G18) | Arc lengths, angles and areas of sectors: additional Foundation content, so it is on Foundation and Higher papers. |
| GCSE Edexcel 1MA1 (G18) | Arc lengths, angles and areas of sectors: Foundation and Higher. |
Quick checklist for the exam
- Write the fraction θ/360 and simplify it.
- Arc: fraction × 2πr. Area: fraction × πr².
- Perimeter? Add the two radii.
- Major sector? Use 360° − θ.
- Give π answers exactly; otherwise 3 significant figures, with units.
Practise this in Area & Volume Workshop
Ten workshop stations, from rectangles and circles to sectors, cones, similar solids and multi-step exam problems. Free, no sign-up.
Frequently asked questions
Is the arc length formula given in the IGCSE exam?
No. The 0580 List of formulas gives the circumference (2πr) and area (πr²) of a circle, but not arc length or sector area. Learn them as fractions of those two formulas.
What does “give your answer in terms of π” mean?
Leave π as a symbol in the answer, for example 12π cm², instead of working out a decimal. These questions are common on the non-calculator Paper 2.
What is the difference between an arc and a chord?
An arc is curved: it is part of the circumference. A chord is a straight line between two points on the circle.
How do I find the perimeter of a sector?
Add the arc length and the two radii: perimeter = θ/360 × 2πr + 2r.
Are sectors on GCSE Foundation?
Yes. Arc length and sector area (G18) are on both Foundation and Higher papers for AQA and Edexcel.
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