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: *L* = ( *θ*/360)×(2π_r_).

## 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:

*c* = 2_r_ sin (*θ*/2)

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

*c*/2_r_ = sin (*θ*/2)

In this example, (*c*/2_r_) = (2/[2 x 5]) = 0.20.

## Find the Inverse Sine of (θ/2)

Since you now have 0.20 = sin (*θ*/2), 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).

sin^{-1}(0.20) = 11.54 = (*θ* /2)

23.08 = *θ*

## Solve for the Arc Length

Going back to the equation *L* = (*θ*/360) × (2π_r_), input the known values:

*L* = (23.08/360) × ( 2π_r_) = (0.0641) × (31.42) = 2.014 meters

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