Measuring angles in radians
A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. A full turn is 2π radians, so 360° = 2π rad and 180° = π rad.
Key idea
To convert: degrees → radians multiply by π/180; radians → degrees multiply by 180/π. For example 60° = 60 × π/180 = π/3 rad.
Arc length and sector area
When the angle θ is measured in radians, the formulae are beautifully simple. For a circle of radius r and a sector angle θ:
Key idea
Arc length s = rθ. Sector area A = ½r2θ.
These only work with radians, so always convert a degree angle first. The perimeter of a sector is the two straight radii plus the arc: P = 2r + rθ.
Worked example
A sector has radius r = 6 cm and angle θ = 0.5 rad. Find the arc length and the sector area.
Arc length s = rθ = 6 × 0.5 = 3 cm.
Sector area A = ½r2θ = ½ × 62 × 0.5 = ½ × 36 × 0.5 = 9 cm2.
Remember
- 1 radian ≈ 57.3°.
- Use radians in s = rθ and A = ½r2θ, never degrees.
- Keep answers in terms of π when an exact value is asked for.