Radian Measure and Arc Length
Convert between degrees and radians and calculate arc length from radian measure.
Covers the unit circle, degree-to-radian conversion, and worked examples.
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
Key Notes
- The arc length formula s = rθ requires θ in radians — if θ is in degrees, convert first; using degrees directly produces a result that is 57× too large
- Nearly all programming language math functions (sin, cos, Math.sin, etc.) expect radians — forgetting to convert is one of the most common bugs in geometry code
- A full wheel rotation gives arc length 2πr (the circumference), showing why radians are the natural unit: the arc length for one full turn equals exactly 2π times the radius
- Angular velocity ω (rad/s) relates to linear speed by v = rω — this comes directly from differentiating s = rθ with respect to time
Key Notes
- Formula: s = rθ (θ in radians): Arc length equals radius times central angle in radians. For a full circle (θ = 2π): s = 2πr — the standard circumference formula falls out naturally. In degrees: s = (θ° / 360°) × 2πr.
- Radians are the natural unit for this formula: One radian is defined as the angle where arc length equals the radius (s = r × 1). This is why calculus uses radians exclusively: d/dθ sin θ = cos θ only when θ is in radians — in degrees, a factor of π/180 appears.
- Converting between radians and degrees: θ_rad = θ_deg × π/180; θ_deg = θ_rad × 180/π. Key values: 30° = π/6; 45° = π/4; 60° = π/3; 90° = π/2; 180° = π; 360° = 2π. Memorizing these stops most conversion errors.
- Angular velocity and arc length rate: If an object moves along a circle of radius r at angular velocity ω (rad/s), its tangential (linear) speed is v = rω — directly from the arc length formula differentiated with respect to time. This is why radians make rotational mechanics so clean.
- Applications: Arc length in radians is used in gear and belt drive design (arc of contact), road curve design (arc length for a given radius and turning angle), angular velocity to linear speed conversion, pendulum arc distance, and the great-circle distance formula on a sphere.