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Arc Length Calculator

Enter a radius and a central angle to get arc length first, plus sector area, chord length, sector perimeter, and the sector share of a full circle. Switch between degrees and radians without converting by hand.

How It Works

Formula

θ=α⋅π180∘\theta = \alpha \cdot \frac{\pi}{180^\circ}

s=rθs = r\theta

A=12r2θA = \frac{1}{2}r^2\theta

c=2rsin⁡(θ2)c = 2r\sin\left(\frac{\theta}{2}\right)

P=s+2rP = s + 2r

share=θ2π⋅100%\text{share} = \frac{\theta}{2\pi}\cdot 100\%

Variables, symbols and units

rr

Radius from the center to the arc

α\alpha

Central angle when the input is given in degrees

θ\theta

Central angle in radians used by the formulas

ss

Arc length, the curved edge

AA

Sector area, the covered region

cc

Chord length, the straight span between the arc endpoints

PP

Sector perimeter, arc plus two radii
Calculation method explained

Choose whether the angle is in degrees or radians, then enter the radius and the central angle for one positive sector. If you use degrees, the calculator first converts that value to radians. From the same radius r and radian angle θ it derives the curved arc length, covered area, straight chord, full sector perimeter, and the sector share of a full circle.

The workflow is deliberately narrow:

  • Convert the angle to radians when needed.
  • Use s=rθs = r\theta for the curved edge.
  • Use the same rr and θ\theta to derive sector area, chord length, and sector perimeter.
  • Keep the interpretation honest: results are geometric estimates from your inputs only.

Use circle for a full turn, triangle when the straight span is the main object, and regular-polygon when repeated equal edges are the better model than a true arc.

Frequently Asked Questions

When should I use degrees instead of radians?
Use the unit your source gives you. Drawings, worksheets, and field notes often use degrees. Trigonometry, calculus, CAD exports, and some textbooks use radians. This calculator accepts either, then runs the same sector formulas after converting degree input to radians.
Why is chord length shorter than arc length?
The chord is the straight line between the arc endpoints. The arc length follows the curve. For any positive sector smaller than a full turn, the straight shortcut is shorter than the curved path.
Why does sector perimeter include two radii?
Sector perimeter is the whole boundary of the slice: one curved arc plus the two straight sides from the center to the arc endpoints. Arc length is only the curved part.
Why does this stop below a full turn?
This tool is intentionally for one sector only, so it accepts angles greater than 0 and smaller than one full turn. If your angle is 360° or 2π, that is whole-circle work and belongs in the circle calculator.
How is this different from circle, triangle, and regular-polygon calculators?
Use circle for full-circle circumference and area, triangle when the straight span or side relationships are the real job, and regular-polygon when you are approximating a curve with repeated equal edges. Arc length is the partial-circle tool for radius plus central angle.

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