In circular measure angles are given in radians. For a sector of radius r and angle θ radians, the arc length and area follow two simple formulas.
The key formulas
Arc length s = rθ and sector area A = ½r²θ, where θ must be in radians. To convert, use π radians = 180°, so 60° = π/3 radians.
Key idea
s = rθ, A = ½r²θ (θ in radians). Perimeter of a sector = 2r + rθ (two radii plus the arc).
Working with sectors
Rearranging these formulas lets you find r or θ from a given length or area. Always check that θ is in radians before substituting.
Worked example
A sector has radius 6 cm and angle π/3 radians. Arc length s = 6 × π/3 = 2π ≈ 6.28 cm. Area A = ½ × 6² × π/3 = ½ × 36 × π/3 = 6π ≈ 18.8 cm².
Combined shapes, such as a sector with a triangle removed, are handled by adding or subtracting the relevant sector areas and lengths. Give final answers to a sensible accuracy.
The area of the triangle formed by two radii is ½r²sinθ, so the area of a circular segment is the sector area minus this triangle, ½r²(θ − sinθ).
Remember
- Convert degrees to radians before using s = rθ or A = ½r²θ.
- The perimeter of a sector includes the two straight radii.