The Length of an Arc

Recall, the circumference of a circle is 2πr. In the diagram below, the arc length l or XY subtends an angle θ° at the centre O with radius r.

In general, the length of an arc of a circle is proportional to the angle at which the arc subtends at the centre.

Therefore, the arc length “l” or XY is given as

l = \( \frac {θ}{360}\scriptsize \: \times \: 2 \pi r \)

Example 1.2.1:

In terms of π, what is the length of an arc which subtends an angle of 30º at the centre of a circle of radius \( \scriptsize 3\normalsize \frac {1}{2} \: \scriptsize cm \)?

Solution:

Using XY = \( \frac {θ}{360}\scriptsize \: \times \: 2 \pi r \)

θ = 30º

radius = \(\scriptsize 3\frac {1}{2} = \normalsize \frac {7}{2}\)