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
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.