Radians, arcs and sectors

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What this lesson covers

  • π radians equals 180°.
  • Arc length is rθ.
  • Sector area is ½r²θ.
Key idea

Measuring in radians

  • π radians = 180°.
  • Multiply by π/180 to reach radians and by 180/π to return.
  • With θ in radians: s = rθ and A = ½r²θ.
Worked example

r = 8, θ = 1.5 radians

  • These are direct substitutions.
  1. s = 8 × 1.5 = 12
  2. A = ½ × 64 × 1.5
  3. A = 48 square units
Worked example

r = 6, angle 120°

  • Convert first.
  1. 120° = 2π/3 radians
  2. s = 6 × 2π/3 = 4π = 12.57
  3. A = ½ × 36 × 2π/3
  4. A = 12π = 37.70 square units
Worked example

A segment

  • r = 10, θ = 1.2. A segment is the sector minus the triangle.
  1. sector = ½ × 100 × 1.2 = 60
  2. triangle = ½ × 100 × sin 1.2 = 46.60
  3. segment = 13.40 square units

Common mistakes

  • Using s = rθ with θ in degrees.
    These formulas only work in radians.
  • Leaving the calculator in degrees for sin 1.2.
    Switch to radians.

Your turn

  • 1. Convert 45° to radians.
  • 2. Convert 2.5 radians to degrees.
  • 3. For r = 5 and θ = 0.8, find the arc length and the sector area.
Show answer

1. π/4 2. 143.2° 3. s = 4, A = 10

Summary

What to take away

  • Convert to radians before using the formulas.
  • Segment = sector − triangle.

More in AS 91259

The sine and cosine rulesNotesArea of any triangleNotesBearings and 3D problemsNotesRadians, arcs and sectorsNotes

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