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MathidiaMathematics Academy
Trigonometry and radians
Lesson 21 of 75
Introduction

Radians and arc geometry

Objective. Convert angle measures and use radian arc and sector formulae.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Radians measure angles using arc length relative to radius, with π radians equal to 180°, so converting uses angle in radians = angle in degrees × π/180. Arc length is given by s=rθ and sector area by A=½r²θ, both requiring θ in radians, not degrees. For a sector with radius 6 cm and angle π/3 radians, the arc length is 6×π/3=2π cm and the area is ½×36×π/3=6π cm². Always confirm θ is in radians before applying either formula, since using degrees gives drastically wrong results.

Worked example

A sector has radius 10 cm and angle 2 radians. Find the exact arc length of the sector.

Diagram of a circle
  1. 1Recall the arc length formula s=rθ
  2. 2Substitute r=10 and θ=2
  3. 3Multiply: s=10×2
  4. 4State the arc length as 20 cm

Answer: 20 cm

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Practice

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Common mistake. Students use arc length formula s=rθ with θ still in degrees, computing s=6×60=360 instead of converting 60° to π/3 radians first, giving the correct s=2π.

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