How to Calculate Arc Length
An arc is a portion of the circumference of a circle. The arc length depends on both the radius of the circle and the central angle that the arc subtends. This concept appears in engineering, architecture, and anywhere circular or curved geometry is involved.
The Formula (Degrees)
Arc Length = (θ / 360) × 2πr
The Formula (Radians)
Arc Length = r × θ
Where θ is the angle in radians and r is the radius. This simpler form is why radians are preferred in advanced math.
Step-by-Step Example
Find the arc length of a 60° arc on a circle with radius 10 cm.
Using degrees:
- Arc Length = (60/360) × 2π × 10
- = (1/6) × 62.832
- = 10.47 cm
Using radians: Convert 60° to radians: 60 × π/180 = π/3 ≈ 1.047 rad Arc Length = 10 × 1.047 = 10.47 cm ✓
Converting Between Degrees and Radians
Radians = Degrees × (π / 180)
Degrees = Radians × (180 / π)
Common Arc Lengths
| Angle | Fraction of Circle | Arc Length (r = 1) |
|---|---|---|
| 30° | 1/12 | 0.524 |
| 45° | 1/8 | 0.785 |
| 90° | 1/4 | 1.571 |
| 180° | 1/2 | 3.14159 |
| 360° | Full circle | 6.283 |
Applications
- Engineering: Calculating the length of curved roads or bent metal
- Watch design: Determining gear tooth arc length
- Robotics: Path planning for circular movements
Use our arc length calculator to compute any arc length from radius and angle.