Arc Length Formula
Reference for arc length L = rθ (radians) and L = (θ/360) × 2πr (degrees).
Covers track design, belt drives, and gear pitch with radian/degree conversion.
Need to calculate, not just reference? Use the interactive version.
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The Formula
s = rθ (radians) or s = (θ/360) × 2πr (degrees)
Arc length is the distance along a curved section of a circle. Using radians makes the formula beautifully simple — just multiply the radius by the angle.
Variables
| Symbol | Meaning |
|---|---|
| s | Arc length (same units as radius) |
| r | Radius of the circle |
| θ | Central angle (in radians or degrees) |
Example 1
Find the arc length for a 60° angle on a circle with radius 10 cm
s = (60/360) × 2π × 10
s = (1/6) × 62.83
s ≈ 10.47 cm
Example 2
An angle of π/4 radians on a circle with radius 8 m
s = r × θ = 8 × π/4
s = 2π ≈ 6.28 m
When to Use It
Use the arc length formula when:
- Calculating the length of curved roads or tracks
- Measuring distances along circular paths
- Designing gears, pulleys, and circular components
- Converting between angle measure and distance along a curve
Key Notes
- The radian formula s = rθ is cleaner than the degree form — this is the main reason radians are preferred in mathematics and physics
- A full circle (θ = 2π rad) gives s = 2πr, which is simply the circumference — confirming the formula is consistent
- Arc length units match the radius units; if r is in centimeters and θ is in radians, s comes out in centimeters
Key Notes
- Formula: s = rθ (θ in radians): Arc length equals radius times central angle in radians. In degrees: s = (θ°/360°) × 2πr. For a full circle (θ = 2π), s = 2πr — the familiar circumference formula is a special case.
- Radian definition is built into this formula: One radian is defined as the angle for which arc length equals the radius (s = r when θ = 1 rad). This is why radians make trig formulas cleaner — no conversion factor needed.
- Chord vs arc: The chord (straight line between two endpoints) is always shorter than the arc: chord = 2r sin(θ/2) ≤ s = rθ. For small angles they converge: as θ → 0, sin(θ/2) ≈ θ/2 and chord ≈ arc.
- Arc length of a curve: s = ∫√(1 + (dy/dx)²) dx: For non-circular curves defined by y = f(x), arc length requires calculus. This integral is derived from the Pythagorean theorem applied to infinitesimal line segments along the curve.
- Applications: Arc length calculations are used in road and railway curve design, belt and chain length around pulleys, clock hand travel distances, CNC toolpath planning, and the haversine formula for great-circle distances on Earth.