Key Idea: A sector is a 'pie slice' of a circle — defined by two radii and an arc. Topic 3.4 is about calculating the length of the arc and the area of the sector using the angle θ (in degrees). These formulas also combine to give the perimeter of a sector and the area of a circular segment.
✅ Core formulas
| Quantity | Formula | Note |
|---|---|---|
| Arc length | l = (θ/360) × 2πr | The fraction of the full circumference |
| Sector area | A = (θ/360) × πr² | The fraction of the full circle area |
| Sector perimeter | P = l + 2r | Arc length plus both radii (the straight edges) |
| Segment area | A = sector area − triangle area | Triangle area = ½r² sinθ (the chord cuts off the segment) |
Example: Arc length: radius = 8 cm, angle = 135°. l = (135/360) × 2π(8) = (3/8) × 16π = 6π ≈ 18.8 cm Sector area: radius = 5 cm, angle = 72°. A = (72/360) × π(5²) = (1/5) × 25π = 5π ≈ 15.7 cm² Segment area: radius = 10 cm, angle = 60°. Sector area = (60/360) × π(100) = 100π/6 ≈ 52.36 cm² Triangle area = ½(10²)sin60° = 50 × (√3/2) ≈ 43.30 cm² Segment area ≈ 52.36 − 43.30 = 9.06 cm²
The angle θ must be in degrees when using these formulas. If you are given radians, convert first: θ (degrees) = θ (radians) × 180/π. A segment is NOT the same as a sector. Sector = pie slice (includes both radii). Segment = region between an arc and its chord (no straight radii visible).
Paper 1 (GDC allowed): Leave answers in terms of π when possible (e.g., 6π cm). This is exact and avoids rounding errors. Paper 2 (GDC allowed): Evaluate to 3 s.f. For segment area questions, calculate sector and triangle areas in sequence and show both — the examiner wants to see both values.
IB-style question [6 marks]
A landscaped flower bed is in the shape of a sector of a circle. It has two straight edges (radii) of length 7 m and a central angle of 120°. (a) Find the length of the curved edge of the flower bed. (b) Find the perimeter of the flower bed. (c) Find the area of the flower bed.
Step by step:
(a) Write the arc-length formula and substitute r = 7 m, θ = 120°.
(b) The perimeter is the arc plus the two straight radii.
(c) Write the sector-area formula. Area uses r², not r.
Substitute r = 7 m and θ = 120°.
Final answer:
(a) Arc = 14π/3 ≈ 14.7 m. (b) Perimeter ≈ 28.7 m. (c) Area = 49π/3 ≈ 51.3 m².