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Arc Length Formula in Radians (L = rθ)

Understand the radian arc length formula L = rθ, convert degrees to radians, and see why this form is used in trigonometry and calculus.

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Blueprint-style circular sector diagram for arc length guides

Quick Answer

L = rθ when θ is in radians. Convert degrees first with θ_rad = θ_deg × (π / 180).

Formula

  • L = rθ
  • θ_rad = θ_deg × (π / 180)
  • Full circle: θ = 2π rad
  • L = 2πr at one revolution

Introduction

Radians measure angle by arc length per unit radius. One radian is the angle where the intercepted arc equals the radius. That definition is why the arc length formula collapses to L = rθ with no extra constants.

Degree measure is still valid everywhere this site discusses practical layout work. Radians become essential when you read trigonometry texts, physics derivations, or calculus problems where θ appears inside sine and cosine functions.

The home calculator accepts degrees and converts internally, so you can stay in the unit printed on your diagram.

Main Content

Why radians simplify the math

In degree form, you scale the angle against 360° before multiplying by circumference. Radians bake that scaling into the angle unit itself, so arc length becomes a direct product of radius and angle.

On the unit circle, an angle of θ radians intercepts an arc of length θ when the radius is 1. Scale up to radius r and the arc scales linearly to rθ. That linearity is what makes radians natural in advanced courses.

If you learned arc length first through degrees, read the degree-based arc length formula for the parallel explanation using (θ/360°) × 2πr. Both describe the same geometry.

Central angle and arc length grow together in radian measure. Doubling θ doubles L when r is fixed, which is the proportional relationship described in our guide on arc length and central angle.

Radian relationship and interpretation

  • L = rθ (θ radians)
  • 1 rad ≈ 57.3°
  • π rad = 180°
  • 2π rad = 360°

Conversion and calculation example

Find arc length for r = 5 m and θ = 60°. Convert: θ_rad = 60 × (π/180) = π/3. Then L = 5 × (π/3) ≈ 5.236 m.

Check with degrees: L = (60/360) × 2π(5) = (1/6) × 10π ≈ 5.236 m. The match confirms the conversion.

For θ = π/2 rad and r = 4 m, no conversion is needed: L = 4 × (π/2) = 2π ≈ 6.283 m.

FAQ

When should I prefer radians?
Use radians in trigonometry, calculus, and physics where formulas already assume radian measure. Use degrees when the problem or drawing gives degrees.
Is L = rθ ever wrong?
It is wrong only if θ is still in degrees. Always convert first.

Conclusion

Radians align angle size with arc distance on the circle. That is why L = rθ is the compact form.

Pick degrees or radians based on the source data, not personal preference, and convert before mixing formulas.

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