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Arc Length and Central Angle Explained

See how central angle and arc length relate on a circle, how to convert angle units, and why arc distance scales with θ at fixed radius.

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Blueprint-style circular sector diagram for arc length guides

Quick Answer

Central angle θ at the center determines what fraction of the circumference the arc spans. L = (θ/360°) × 2πr in degrees, or L = rθ in radians.

Formula

  • L ∝ θ when r is fixed
  • 360° → full circumference
  • 180° → semicircle arc πr
  • 90° → quarter circumference

Introduction

Central angle and arc length scale together when radius stays fixed. Double the angle and the curved distance doubles. Halve the radius and both arc and chord shrink proportionally.

Understanding that proportion prevents guesswork on multiple-choice geometry questions and on site when only angle size changes between two similar bends.

Use the calculator to test several angle values on the same radius and watch arc length respond linearly.

Main Content

Central angle defined and applied

The central angle has its vertex at the circle center. Its sides are radii that meet the arc at the endpoints. Any angle measured on the circumference is an inscribed angle unless the problem converts it.

In degree measure, a 360° central angle produces arc length equal to the full circumference 2πr. A 180° angle gives a semicircular arc of πr. Each proportional step down reduces arc length linearly.

Radians express the same idea directly: θ rad intercepts arc length rθ. That is why radian measure appears throughout trigonometry once arc length becomes a routine calculation.

If the vocabulary is new, read what is arc length first, then return here for the angle relationship.

When you need the numeric formula with degrees, the arc length formula article shows substitution patterns with worked structure.

Same radius, different angles

Let r = 4 m. At θ = 90°, L ≈ 6.283 m. At θ = 180°, L ≈ 12.566 m, exactly double. At θ = 45°, L ≈ 3.142 m, half of the 90° case.

This table behavior is useful when a machine setting changes only the sweep angle while the bend radius stays constant.

FAQ

Can central angle exceed 360°?
In pure geometry problems it sometimes does to represent more than one revolution. Arc length scales beyond 2πr accordingly.
Is inscribed angle ever equal to central angle?
An inscribed angle that intercepts the same arc is half the central angle when the vertex sits on the major arc side. Always read the diagram carefully.

Conclusion

Central angle is the control variable for arc length at fixed radius. Learn to spot it on the diagram before substituting.

Linear scaling with angle is one of the fastest sanity checks available.

Calculate for your angle