Area of a sector
A sector is the region enclosed by two radii and an arc. When the angle θ at the centre is measured in radians, the area of the sector is a fraction of the whole circle. Since a full circle of area πr2 corresponds to an angle of 2π, a sector of angle θ has area A = ½r2θ. An equivalent form using the arc length s is A = ½rs.
Key formula
A = ½r2θ (θ in radians) · also A = ½rs · segment area = ½r2(θ − sin θ)
Area of a segment
A segment is the region between a chord and an arc. Its area is found by subtracting the area of the triangle from the area of the sector: area of segment = ½r2θ − ½r2sin θ = ½r2(θ − sin θ). When using sin θ, make sure the calculator is in radian mode.
Worked example
A sector has radius 6 cm and angle 1.5 rad. Its area is A = ½ × 62 × 1.5 = ½ × 36 × 1.5 = 27 cm2. For a segment with radius 10 cm and θ = 1.2 rad, area = ½ × 100 × (1.2 − sin 1.2) = 50 × (1.2 − 0.932) ≈ 13.40 cm2.
Finding r or θ
The formula rearranges to θ = 2A/r2 and r = √(2A/θ). Always substitute carefully and keep the angle in radians throughout.
Remember
- Sector area A = ½r2θ needs θ in radians.
- Segment area = sector area − triangle area.
- Triangle area in a sector = ½r2sin θ.