Arc Length CalculatorFree online math calculator — instant results, no sign-up
Calculate circular arc length from radius and central angle.
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About the Arc Length Calculator
The Arc Length Calculator finds the distance along a circular arc using a radius and a central angle entered in degrees. It converts the angle to a fraction of a full turn and applies that fraction to the circle’s circumference. With the default radius of 5 and central angle of 60 degrees, the arc length is 5.235987756 in the same length unit as the radius.
- Radius
- The radius is the distance from the circle’s center to its edge, entered in a length unit such as centimeters or meters, with a default of 5.
- Central angle (degrees)
- The central angle is the angle at the circle’s center that spans the desired arc, entered in degrees with a default of 60.
Formula and method
s = 2πr × (θ / 360) = πrθ / 180; with r = 5 and θ = 60, s = π × 5 × 60 / 180 = 5π / 3 ≈ 5.235987756.
- sArc length, measured in the same length unit as the radius.
- rRadius of the circle.
- θCentral angle expressed in degrees.
- πPi, the ratio of a circle’s circumference to its diameter.
Divide the central angle by 360 to determine the fraction of the circle covered by the arc, then multiply by the circumference, 2πr. For the default inputs, 60 / 360 = 1 / 6 and the circumference is 10π, so the arc length is 10π / 6 = 5π / 3, displayed as 5.235987756.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Radius5
- Central angle (degrees)60
Results
- Result5.235987756
How to use the Arc Length Calculator
- 1Enter the circle’s radius in your chosen length unit; the default is 5.
- 2Enter the central angle in degrees; the default is 60.
- 3Check that the angle describes the arc you want, rather than the angle for the remaining part of the circle.
- 4Calculate the arc length using s = πrθ / 180.
- 5Read the Result in the same unit as the radius; the default inputs give 5.235987756.
Reading your result
The Result row gives the distance along the circle’s curved edge between the arc’s endpoints, not the straight-line distance between them. For radius 5 and central angle 60 degrees, Result = 5.235987756. If the radius is in centimeters, this result is in centimeters; if it is in meters, the result is in meters.
Limitations
The formula assumes a circular arc with a constant radius and does not apply directly to ellipses or irregular curves. Use a positive radius for a nondegenerate circle and a nonnegative angle for an ordinary arc length. Angles from 0 to 360 degrees describe up to one complete circle; larger angles mathematically represent travel through more than one turn. Angles must be entered in degrees, and the displayed decimal result is rounded.
Common questions
Can I enter the central angle in radians?+
This calculator expects degrees. Convert radians to degrees by multiplying by 180 / π before entering the angle; the equivalent arc-length formula for a radian angle is s = rθ.
Does the result measure the arc or the chord?+
It measures the curved arc. The chord is the straight segment joining the endpoints and is calculated with 2r sin(θ / 2), using degrees for the sine when θ is in degrees.
How do I calculate the longer arc between two endpoints?+
If your given angle describes the shorter arc, enter 360 degrees minus that angle to calculate the longer arc. The radius stays the same.
Why do the default inputs produce 5.235987756?+
A 60-degree angle covers one-sixth of a circle. With radius 5, the circumference is 10π, so the arc length is one-sixth of 10π, or 5π / 3, which rounds to 5.235987756.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
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Results are estimates for informational purposes and are not financial, medical or legal advice.