Arc Length Calculator

Arc length is the curved distance along part of a circle's edge. It shows up in geometry homework, circular motion problems, gear design, road layout, and surveying. Give this calculator any two values among central angle, radius, arc length, and sector area, and it fills in the rest, along with diameter and chord.

What is arc length?

An arc is a slice of a circle's circumference. Its length depends on two things: how big the circle is, and how much of it the slice covers. In symbols:

L = \theta r

with $\theta$ in radians. If you'd rather think in degrees, multiply by $\pi/180$ :

L = \frac{\pi r \theta}{180}

Take a circle with radius 5 m and a 60° central angle. The arc length works out to L = 5 × (60π/180) ≈ 5.236 m, and the sector (the pizza slice bounded by the arc and the two radii) has area $A = \frac{1}{2} \times 5.236 \times 5 \approx 13.09 \text{ m}^2$ . Doubling the radius doubles the arc length, and doubling the angle does the same.

How to use this calculator

Type any two known values into the inputs and pick the units you want: degrees or radians for the angle, metric or US units for the lengths. The remaining fields fill in as you type. The order you enter them doesn't matter. Radius and angle works, angle and arc length works, sector area and radius works too.

Continuing the 5 m / 60° example: those two inputs alone are enough to recover the arc length, sector area, diameter, and chord.

Where this comes up

In civil engineering, road and railway designers use arc length to plan turn radii and bridge spans. Mechanical engineers reach for it when cutting gear teeth, laying out cams, or sizing pulley belts. Physicists multiply angular displacement by radius to get the linear distance traveled by a point on a spinning object. Surveyors and navigators use great-circle arcs to measure distances across Earth's surface, since straight lines don't fit a sphere. And in sheet-metal work, bending material around a die needs the unfolded arc length so the part cuts to the right size.

Same relationship, rearranged

Two equations do all the work, and the calculator rearranges them based on what you give it.

When you know the angle and radius:

$L = \theta r$ , arc length from angle and radius
$r = L / \theta$ , radius from arc length and angle
$\theta = L / r$ , angle from arc length and radius

When the sector area is in the mix:

$A = \tfrac{1}{2} L r$
$L = 2A / r$
$r = 2A / L$

Two helpers round things out. The diameter is just twice the radius, $d = 2r$ . The chord, a straight line between the arc's endpoints, follows $c = 2r \sin(\theta/2)$ . Angles inside these formulas are always in radians; the calculator converts for you when you enter degrees.

Frequently asked questions

What's the difference between arc length and chord?

The arc follows the curve; the chord cuts straight across between the same two endpoints. For a 60° arc on a 5 m radius, the arc measures 5.236 m and the chord measures 5 m. As the angle gets smaller, the two values converge, and at tiny angles they're essentially the same length.

Degrees or radians, which should I use?

Whichever you have. Radians keep formulas clean (no π/180 floating around), so physics and calculus problems usually live in radians. Most everyday geometry, from protractors to compass bearings, sticks with degrees. The unit selector swaps between them, so use what you're comfortable with.

Can the central angle go above 360°?

Not for a single arc, since 360° is one full circle. Anything beyond that wraps you around again, which is a different problem (think coiled springs or helical paths). The calculator caps the input at 2π radians for that reason.

My chord came out longer than the arc. What went wrong?

Almost certainly a unit mix-up. The chord is always shorter than the arc for any angle between 0 and 2π. If you see the opposite, the angle is probably entered in degrees but interpreted as radians somewhere, or the radius units don't match the arc length units. Double-check the unit selectors on each field.