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Arc Length Formula (Degrees)

Use the degree arc length formula L = (θ/360°) × 2πr. Step-by-step explanation, radius relationships, and circle geometry for accurate results.

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Blueprint-style circular sector diagram for arc length guides

Quick Answer

L = (θ / 360°) × 2πr where θ is the central angle in degrees and r is the radius in your chosen linear unit.

Formula

  • L = (θ / 360°) × 2πr
  • L = (θ / 360°) × πd
  • L = 2A / r (from sector area)
  • C = 2πr (full circumference)

Introduction

The standard degree formula expresses arc length as a fraction of the full circumference. The fraction equals the central angle divided by 360°, because a full circle spans 360° by definition.

Before substituting numbers, confirm you are measuring the central angle at the circle center. If your diagram labels diameter instead of radius, convert with r = d/2 or use the diameter form of the same relationship.

Open the Arc Length Calculator when you have numeric values ready. The sections below explain why the formula works and when to use alternate forms.

Main Content

Why the formula has that shape

Circumference is the arc length for a full 360° turn. Any smaller central angle takes only a proportional slice of that distance. Multiplying the fraction θ/360 by 2πr therefore gives the curved distance for that slice.

When problems move into trigonometry or calculus, the same relationship is often written with radians as L = rθ. That shorter form is equivalent after you convert degrees with θ_rad = θ_deg × (π/180). Our article on the arc length formula in radians explains why radians remove the conversion factor from the expression.

If a plan gives sector area A instead of angle, you can still reach arc length through L = 2A/r without finding θ first. That shortcut is especially useful when a shaded region is labeled but the angle is not printed.

Formula explanation and circle geometry

  • L = (θ / 360°) × 2πr
  • Circumference C = 2πr
  • Arc fraction = θ / 360°
  • L = (θ / 360°) × C

Applying the formula

  1. Write the angle in degrees Use the central angle at the circle center. Inscribed angles on the circumference require a different relationship unless the problem converts them for you.
  2. Confirm the radius unit Radius, arc length, and chord length must share the same linear unit. Convert millimeters to meters before mixing values.
  3. Substitute and compute Multiply (θ/360) by 2πr. Leave π in symbolic answers when an exact value is required.
  4. Cross-check the result Compare your answer to the step-by-step workflow in how to calculate arc length if the number seems too large or too small for the diagram.

FAQ

Can I use diameter directly?
Yes. Replace r with d/2, or use L = (θ/360°) × πd without finding radius first.
What if θ is in radians?
Use L = rθ instead. Do not plug radians into the degree formula without converting.

Conclusion

The degree formula is the most common entry point for school and field work. It scales any central angle against the full circumference.

Convert to radians only when the problem statement or course expects L = rθ. Both paths give the same curved distance when θ is converted correctly.

Try the formula in the calculator