Arc Length Calculator

The arc length calculator uses s = rθ: the radius times the central angle in radians. Enter any two of radius, angle (in degrees) and arc length to find the third; a 10 cm radius at 90° gives 15.71 cm.

Sources: OpenStax Algebra and Trigonometry 2e, Wikipedia Updated October 16, 2026

Enter exactly two values to calculate the third:

Radius (r):

Central Angle (θ in degrees):

Arc Length (s):

Blue: Radius (r) | Green: Angle (θ) | Red: Arc (s)

Arc length formula

s = r × θ (in radians) = r × (θ° × π/180)

Where: s = arc length, r = radius, θ = central angle

Which value the arc length calculator solves for

The arc length formula s = rθ has three quantities: arc length s, radius r and central angle θ in radians. Fill in any two and the calculator works out the third, choosing the rearranged formula from the empty box.

If only one box, or all three, are filled, it asks you to correct the input. One value is not enough to solve, and three may contradict each other.

How to calculate arc length from the radius and angle

Convert the angle to radians: 90° × π ÷ 180 = 1.5708 radians.
Multiply by the radius: s = 10 × 1.5708 = 15.71 cm.
Check against the whole circle: the circumference is 2π × 10 = 62.83 cm, and a quarter of that is 15.71 cm.

The radian conversion is the step most often missed. OpenStax Algebra and Trigonometry defines the radian as the angle whose arc equals the radius, which is why s = rθ needs no extra constant. The full-circle check uses the circle circumference calculator.

Circle with a 90° sector shaded and its arc drawn thick. Radius 10 cm, angle 1.5708 radians, arc length 15.71 cm, a quarter of the 62.83 cm circumference. — A 90° angle takes a quarter of the circle, so the arc is a quarter of the 62.83 cm circumference.

Finding the angle or the radius from the arc length

With a radius of 10 and an arc of 15.71, the calculator works out θ = s ÷ r = 1.5708 radians and turns that back into 90°. With an arc of 15.71 and an angle of 90°, it gives r = s ÷ θ = 10.

To find a radius from a circle's area or circumference instead, use the radius calculator.

Reading the arc length diagram

The circle under the inputs redraws as you type, with the radius, the swept angle and the arc in different colours. It is a visual check. The numbers in the solution panel are the exact values.

Where arc length comes up

Arc length is used to cut curved material to length, to lay out curves on roads and railways, and in any problem with a sector or a circular path. For the other measurements of the same circle, the circle calculator gives radius, diameter, circumference and area together.

The angle box always takes degrees. Radius and arc length must be in the same unit, because the tool does not convert units. More circle and solid tools are on the geometry calculators page.