Arc Length Calculator

What this calculator does

An arc is part of the way around a circle, not the whole trip. The curved edge of a slice of pie. A bend in a running track. The sweep a windscreen wiper traces. The arc length is how long that curved path would be if you straightened it out into a line.

Give the tool the radius of the circle and the angle the arc covers, and it measures the curve for you.

Using the calculator

1. Type the radius and pick metres or centimetres.
2. Type the angle in degrees, anywhere from 0 to 360.
3. Press Calculate.

The arc length comes back in the same unit as the radius. An angle of 360 degrees is one full turn, so at 360 the arc is the entire way around the circle.

The formula | arc length = (angle ÷ 360) × 2 × π × radius

The arc length is:

arc length = (angle ÷ 360) × 2 × π × radius

The idea behind it is simpler than the formula looks. An arc is just a fraction of the full circle's edge. A whole circle is 360 degrees, and its edge, the circumference, is 2 × π × radius. So you work out what fraction of the circle your angle covers, which is angle ÷ 360, and take that share of the circumference.

A half circle, 180 degrees, gives half the circumference. A quarter, 90 degrees, gives a quarter. The angle is just telling you how big a bite to take.

Degrees, radians, and the easy mix-up

This tool takes the angle in degrees, which is what most people reach for. There is a second way the same formula is often written, using radians instead:

arc length = radius × angle (with the angle in radians)

It looks almost too short, and that is the whole point of radians. A radian is defined so that an angle of exactly one radian sweeps an arc the same length as the radius. In radians the bookkeeping disappears and you just multiply.

Here is the slip to watch for. Degrees and radians are not interchangeable. One radian is about 57.3 degrees, so dropping a degrees number into the short radians formula by mistake makes the arc come out roughly 57 times too long. To turn a degrees angle into radians, multiply it by π and divide by 180.

What unit the answer comes in

Arc length is a distance, so it lands in the same unit as the radius, metres or centimetres here, and never in square units. The answer is rounded to three decimal places.

A worked example | 72 degrees on a 3 m radius

Say the angle is 72 degrees and the radius is 3 m.

1. Find the fraction of the circle: 72 ÷ 360 = 0.2, which is one fifth.
2. Find the full circumference: 2 × π × 3 = 18.8496 m.
3. Take one fifth of it: 18.8496 × 0.2 = 3.77 m.

So the arc length is about 3.770 m, one fifth of the way around a circle of that size.

From a slice to the whole circle

It helps to keep the ends of the range in mind. At 360 degrees the arc is the whole circumference, the entire trip around. At 180 degrees it is exactly half. At 0 there is no arc at all. Everything in between is just the matching fraction.

If all you have is the diameter rather than the radius, halve it first: radius = diameter ÷ 2. And if you want the whole way around instead of a slice, the circumference of circle calculator does that, while the sector area calculator gives the area of the slice, not just its curved edge.

Questions people ask

What is the arc length for a 90 degree angle?

A quarter of the circumference, since 90 is a quarter of 360. For a radius r that works out to (90 ÷ 360) × 2 × π × r.

Does this use degrees or radians?

Degrees. In radians the formula is the shorter arc length = radius × angle, where one radian is about 57.3 degrees.

What is the difference between arc length and chord length?

Arc length is the curved distance along the edge. The chord is the straight line between the two ends of the arc. The arc is always the longer of the two.

How is arc length related to circumference?

It is a fraction of it, set by the angle. A full circle, 360 degrees, gives the whole circumference.

Can the angle be more than 360 degrees?

This tool keeps it to a single turn, 0 to 360, since past 360 you are simply going around the same circle again.

References

A note on where this comes from. The formula is just the circumference, 2 × π × radius, scaled by the fraction of the circle the angle covers. The value of π the tool uses follows the definition tabulated by the US National Institute of Standards and Technology. The cleaner radians form, arc length = radius × angle, comes straight from the definition of the radian, the angle at which the arc equals the radius. For further reading, see Arc length.

1. National Institute of Standards and Technology (NIST), Digital Library of Mathematical Functions, value and definition of π. https://dlmf.nist.gov/
2. The radian, the angle that subtends an arc equal in length to the radius, as defined in the International System of Units (SI).