Arc Length Calculator

How to calculate the Arc Length?

The arc length of a circle is the distance along the curved part of the circumference between two points on the circle (the length of the "arc" itself, not a straight chord). It depends on:

The radius (r) of the circle

The central angle (θ) that subtends the arc (the angle at the center of the circle)

Formulas

There are two main versions depending on the unit of the angle:

When the central angle is in radians (most precise and common in advanced math/physics):

s = r θ

Where:

s = arc length

r = radius

θ = central angle in radians

When the central angle is in degrees (most common in school/everyday problems):

s = (θ / 360) × 2πr

Or equivalently:

s = (θ × 2πr) / 360

Or simplified: s = (θ × πr) / 180

(This is just the proportion of the full circumference 2πr that corresponds to θ degrees out of 360°.)

Quick conversion note: To switch degrees to radians: θ (radians) = θ (degrees) × (π / 180)

Examples Calculations

Example 1 – Degrees (pizza slice arc)

Radius r = 12 cm

Central angle θ = 60° s = (60 / 360) × 2π × 12

s = (1/6) × 24π

s = 4π

s ≈ 12.57 cm(Alternative: s = (60 × π × 12) / 180 = (720π) / 180 = 4π ≈ 12.57 cm)

Example 2 – Radians (preferred in calculus/physics)

Radius r = 5 m

Central angle θ = 1.5 radians s = r × θ

s = 5 × 1.5

s = 7.5 m

Example 3 – Degrees with a larger angle

Radius r = 8 inches

Central angle θ = 120° s = (120 / 360) × 2π × 8

s = (1/3) × 16π

s ≈ (1/3) × 50.2655

s ≈ 16.76 inches