Arc length

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˚.

Arc with 3cm radius and 144 degree angle

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 of a circle with a 150° angle and 4cm radius

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.)