The *arc length* of a circle is the distance along the outside of that circle between two specified points. If you were to walk one-fourth of the way around a large circle and you knew the circle's circumference, the arc length of the section you walked would simply be the circumference of the circle, 2π*r*, divided by four. The straight-line distance across the circle between those points, meanwhile, is called a chord.

If you know the measure of the central angle *θ*, which is the angle between the lines originating at the center of the circle and connecting to the ends of the arc, you can easily calculate the arc length:

## The Arc Length With No Angle

Sometimes, however, you are not given *θ* . But if you know the length of the associated chord *c*, you can calculate the arc length even without this information, using the following formula:

The steps below assume a circle with a radius of 5 meters and a chord of 2 meters.

## Solve the Chord Equation for θ

Divide each side by 2*r* (which equals the diameter of the circle). This gives

In this example

## Find the Inverse Sine of (θ/2)

Since you now have

you must find the angle that yields this sine value.

Use your calculator's ARCSIN function, often labeled SIN^{-1}, to do this, or refer too the Rapid Tables calculator (see Resources).

## Solve for the Arc Length

Going back to the equation

input the known values:

Note that for relatively short arc lengths, the chord length will be very close to the arc length, as a visual inspection suggests.