Mathematics, Statistics & Geometry

Arc Length Calculator

Calculate arc length from radius and central angle, with sector area, chord length, radian conversion, and step-by-step formula output.

Arc Length
7.854
Sector Area39.27
Chord Length7.654
Angle in Radians0.785
Calculation StepsRadius (r) = 10 Angle (θ) = 45° = 45 * π / 180 = 0.785398 rad Arc Length (s) = r * θ s = 10 * 0.785398 = 7.853982 Area = 0.5 * r² * θ = 39.269908 Chord = 2r * sin(θ/2) = 7.653669

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Arc length formula

Arc length is the distance along the curved edge of a circle sector. The core formula is:

For full-circle distance, use the circumference calculator. If the problem involves a right triangle instead of a circular sector, use the right triangle trigonometry calculator.

s=rθ\begin{aligned} s = r\theta \end{aligned}

Where:
s=
The distance along the curved edge of the circle
r=
The distance from the center to the edge
θ\theta=
The angle subtended at the center (in radians)

The angle must be in radians for the formula s = r * theta. If your angle is in degrees, the calculator converts it first.

How to calculate arc length step-by-step

  1. Identify the radius r.
  2. Identify the central angle.
  3. Convert degrees to radians using theta = degrees * pi / 180.
  4. Multiply radius by angle in radians: s = r * theta.

Example with radius 15 and angle 45 degrees:

theta = 45 * pi / 180 = 0.7854 radians

s = 15 * 0.7854 = 11.781

Sector area and chord length

This calculator also returns:

  • Sector area: A = 0.5 * r^2 * theta
  • Chord length: c = 2r * sin(theta / 2)

Arc length is useful for circle geometry homework, curved-road estimates, pulley and belt layouts, circular windows, and design patterns.

Frequently Asked Questions

To find the arc length, multiply the radius of the circle by the central angle in radians. If your angle is in degrees, first convert it to radians by multiplying by π/180.

Arc length is the linear distance along the curve (the 'crust' of the pizza slice). Sector area is the two-dimensional space inside the slice (the 'cheese' and 'toppings').

While this calculator focuses on circular arcs (the most common requirement), the arc length of any smooth curve can be calculated using calculus by integrating the square root of 1 + (f'(x))² over the desired interval.

A radian is a unit of angular measure based on the radius of a circle. One radian is the angle created when the arc length equals the radius. There are exactly 2π radians in a full circle (360°).

The circumference of a circle is just a special case of arc length where the angle is a full 360° (2π radians). Thus, Circumference = r * 2π = 2πr.

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