Radian Measure and Arc Length
Convert between degrees and radians, and calculate arc length using radian measure.
The natural unit of angle measurement.
The Formula
radians = degrees × (π/180)
degrees = radians × (180/π)
Arc length: s = rθ (θ in radians)
degrees = radians × (180/π)
Arc length: s = rθ (θ in radians)
Radians measure angles based on the radius of a circle. One radian is the angle where the arc length equals the radius. A full circle is 2π radians.
Variables
| Symbol | Meaning |
|---|---|
| θ | Angle (in radians or degrees) |
| π | Pi (approximately 3.14159) |
| s | Arc length |
| r | Radius of the circle |
Common conversions: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π
Example 1
Convert 150° to radians
radians = 150 × (π/180) = 150π/180
= 5π/6 ≈ 2.618 radians
Example 2
A wheel of radius 0.4 m rotates through 3 radians. How far does a point on the rim travel?
s = rθ = 0.4 × 3
s = 1.2 m
When to Use It
Use radian measure when:
- Working with trigonometric functions in calculus
- Calculating arc length and sector area
- Programming — most math libraries expect radians, not degrees
- Analyzing rotational motion in physics and engineering