Arcs and sectors
Working with parts of a circle.
Review these first
Learning objectives
What you need to be able to do
Teacher-mapped phrasing — check against the official Cambridge syllabus for exact wording.
- 5.2.1Calculate arc length, sector area and the perimeter of a sector.
5 minute read
Arcs and sectors
A sector is a "pizza slice" of a circle, bounded by two radii and an arc. An arc is the curved part of the circumference.
The single idea
A sector with angle θ is the fraction θ/360 of the whole circle. Every formula follows:
- Arc length =
(θ/360) × 2πr - Sector area =
(θ/360) × πr²
Rather than memorising two formulas, remember one principle — take that fraction of the whole — and the rest is the circle formulas you already know.
Perimeter of a sector — the classic trap
The perimeter is not just the arc. It is the arc plus the two radii: Perimeter = arc length + 2r Forgetting the two straight edges is the most common error in this topic, because "perimeter" is loosely associated with "the curved bit".
Segments
A segment is cut off by a chord rather than by two radii. Its area is found by: segment area = sector area − triangle area where the triangle is formed by the two radii and the chord, with area ½ab sin C.
Semicircles and quarter circles
These are simply sectors with θ = 180° or 90°. A semicircle's perimeter is πr + 2r, not πr.
Think of it like this
A sector is a slice of pie. The crust is the arc, but the perimeter of the slice includes the two straight cut edges — which is exactly what people forget when they only look at the crust.
Worked examples
Method, step by step
A sector has radius 6 cm and angle 60°. Find its arc length and its perimeter, to 3 significant figures.
- 1The fraction of the circle is 60/360 = 1/6.
- 2Arc length = (1/6) × 2π(6) = (1/6) × 12π = 2π ≈ 6.283 cm.
- 3Perimeter = arc + 2r = 6.283 + 12 = 18.283.
- 4To 3 s.f. that is 18.3 cm.
Arc ≈ 6.28 cm; perimeter ≈ 18.3 cm
Common misconceptions
- Giving the perimeter of a sector as the arc length alone, omitting `+ 2r`.
- Using `θ/180` instead of `θ/360`. The full circle is 360°.
- Confusing a segment with a sector. A sector is bounded by two radii; a segment by a chord.
In the exam
- Write `θ/360` first and multiply it by whichever whole-circle formula you need — one principle covers both arc and area.
- Read carefully for "perimeter" versus "arc length"; they differ by 2r and the question is often testing exactly that.