Arcs and sectors
This is examined on Higher Papers 1H and 2H of Edexcel 1MA1, and occasionally on Foundation 2F (basic application). The skill: scale the formulae for circumference and area by the fraction of the circle the sector occupies.
📖Definition— Definitions
A sector is a "pizza slice" — a region bounded by two radii and an arc. The angle at the centre is θ.
An arc is the curved part of the sector boundary.
Arc length
Arc length = (θ / 360) × 2πr
Sector area
Sector area = (θ / 360) × πr²
Both formulae are the full circle's circumference / area scaled by θ/360.
Common conversions
| θ | Fraction | Arc length | Sector area |
|---|---|---|---|
| 90° | 1/4 | (1/4)(2πr) = πr/2 | (1/4)(πr²) |
| 180° | 1/2 | πr | πr²/2 |
| 60° | 1/6 | (1/3)πr | (1/6)πr² |
| 45° | 1/8 | πr/4 | πr²/8 |
Perimeter of a sector
Perimeter = arc length + 2 × radius (one full slice has two straight edges and one curved).
Reverse problems
Given an arc length and a radius, find θ:
θ = (arc length / (2πr)) × 360°.
Common Edexcel mark-scheme phrasing
- M1 for the correct fraction (θ/360).
- M1 for substituting r and θ correctly.
- A1 for the arc length or area, often to 3 s.f.
- M1 + A1 for the perimeter (arc + 2r).
⚠Common mistakes— Common errors
- Forgetting to add 2r when asked for perimeter (arc length only is wrong).
- Mixing up arc length with sector area formula.
- Using θ in radians (not on Edexcel GCSE — always degrees).
- Stating answer without units, or wrong units (e.g. cm² for length).
AI-generated · claude-opus-4-7 · v3-edexcel-maths-leaves