Arc Length & Sector Area
Taking a fraction of the circumference or of the area.
The explanation
A sector is a slice of a circle. An arc is the curved edge of that slice. Both are just fractions of the whole.
The fraction is θ/360, where θ is the central angle.
Arc length = (θ/360) × 2πr
Sector area = (θ/360) × πr²
A 90° sector is a quarter of the circle: a quarter of the circumference and a quarter of the area.
Watch the units. Arc length is a distance (cm), sector area is a space (cm²). If your answer has an r² in it, it is an area.
A segment — different from a sector — is the region between a chord and its arc. Find it by taking the sector and subtracting the triangle formed by the two radii and the chord.
Arc length and sector area are proportional parts of the whole circle, with proportion θ/360 in degrees:
s = (θ/360)·2πr A_sector = (θ/360)·πr²
In radians both simplify, since θ_rad = 2π(θ/360):
s = rθ A_sector = ½r²θ
The disappearance of the conversion constant is the practical reason radians are preferred in analytic work.
A circular segment is the region bounded by a chord and its arc, computed as the sector minus the triangle: A = ½r²(θ − sin θ) in radians. That subtraction is the step most often omitted.
Dimensionally, arc length is degree 1 in r and sector area degree 2, so scaling a circle by k scales arc lengths by k and sector areas by k² — the same dimensional rule as for similar figures generally.
Answers are normally left in terms of π unless a decimal is requested, since 6π is exact where 18.85 is not.
Worked example
A circle has radius 9. Find the arc length and sector area for a 120° central angle, in terms of π.
- Fraction: 120/360 = 1/3.
- Circumference = 2π(9) = 18π, so arc = 18π/3 = 6π.
- Area = π(81) = 81π.
- Sector = 81π/3.
Answer: Arc length 6π, sector area 27π
Common mistakes
- Using the same formula for both, forgetting the area needs r².
- Computing a segment as the sector without subtracting the triangle.