How To Calculate Arc Lengths Without Angles

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:

\(L = \frac{θ}{360} × 2πr\)

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:

\(c = 2r \sin \bigg(\frac{θ}{2}\bigg)\)

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

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

\(\frac{c}{2r} = \sin \bigg(\frac{θ}{2}\bigg)\)

In this example

\(\frac{c}{2r} = \frac{2}{2×5} = 0.2\)

Since you now have

\(0.2 = \sin \bigg(\frac{θ}{2}\bigg)\)

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).

\(\sin^{-1}(0.2) = 11.54=\frac{θ}{2} \
\implies θ=23.08\)

Going back to the equation

\(L = \frac{θ}{360} × 2πr\)

input the known values:

\(L = \frac{23.08}{360} × 2π × 5\text{ meters}
\ \,
\= 0.0641 × 31.42 = 2.014 \text{ meters}\)

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