Radians

Radians measure angles using the radius: one radian is the angle where the arc is as long as the radius. A full turn is 2π radians, so 180° = π. Calculus with trig only works neatly in radians.

Key facts

  • π rad = 180°.
  • Degrees to radians: × π/180. Radians to degrees: × 180/π.
  • Arc length = rθ; sector area = ½r²θ (θ in radians).
  • Small angles: sin θ ≈ θ, cos θ ≈ 1 − θ²/2.

Worked examples

Example 1

Convert 135° to radians.

  1. 135 × π/180 = 3π/4

Answer: 3π/4

Example 2

A sector has radius 5 cm and angle 1.2 rad. Find its area and arc length.

  1. Area: ½ × 25 × 1.2 = 15 cm²
  2. Arc: 5 × 1.2 = 6 cm

Answer: 15 cm² and 6 cm