Understanding Arc Length
An arc is a portion of the circumference of a circle. The arc length is the distance along the curved line forming the arc, depending on the radius and the central angle.
Arc Length Formula
L = rθ (angle in radians)
L = (θ / 360) × 2πr (angle in degrees)
L = (θ / 360) × 2πr (angle in degrees)
Radian Measure
| Degrees | Radians | Fraction of Circle |
|---|---|---|
| 30° | π/6 | 1/12 |
| 45° | π/4 | 1/8 |
| 60° | π/3 | 1/6 |
| 90° | π/2 | 1/4 |
| 180° | π | 1/2 |
| 360° | 2π | 1 (full) |
Sector Area
A = (1/2)r²θ (radians)
A = (θ/360) × πr² (degrees)
A = (θ/360) × πr² (degrees)
Chord Length
Chord = 2r × sin(θ/2)
Applications
- Engineering: Belt lengths, gear profiles, curved structural elements.
- Geography: Great circle distances using Earth's radius.
- Architecture: Designing arches, domes, and curved walls.
- Physics: Circular motion distance (s = rθ).
- Astronomy: Converting angular sizes to physical sizes.
Arc Length in Calculus
L = ∫ab sqrt(1 + [f'(x)]²) dx
The circular arc length L = rθ is a special case of this general formula.