Arc length is a fraction of circumference. Area of a sector is a fractions of the area of a circle. Both can be calculated using the angle at the centre and the diameter or radius.

Part of

The circumference of a circle = \(\pi d\) or \(2\pi r\).

Look at the sector of the circle shown below. To calculate the length of the arc, we need to know what fraction of the circle is shown. To do this, we use the angle and compare it with 360˚.

This angle is 144°.

That is \(\frac{{144^\circ }}{{360^\circ }} = \frac{2}{5}\) of a full turn (360°).

So the arc is \(\frac{2}{5}\) of the circumference.

\(c=\pi d=3.14\times 6\) (Remember the diameter is double the radius.)

\[=18.84cm\]

Arc length = \(\frac{2}{5}\times 18.84 = 7.54cm\)

(There is so need to simplify \(\frac{144}{360}\), you can use this in the arc calculation instead of \(\frac{2}{5}\).)

The formula used to calculate the Arc Length is:

\[Arc\,length = \frac{{Angle}}{{360^\circ }} \times \pi d\]

Now try the example question below.

- Question
Calculate the length of the arc shown in the diagram below.

\[Arc\,length = \frac{{Angle}}{{360^\circ }} \times \pi d\]

Remember also that \(d = 2 \times r\)

\[= \frac{{150}}{{360}} \times \pi \times 8\]

\[= 10.47cm\]

\[= 7.5\,(to\,1\,d.p.)\]