Length of circular arc (degrees)

The diagram below shows a sector of a circle. It has been formed by an angle of size $\theta$θ centred at the origin and has an arc length (the curved part of the perimeter) of length $l$l.

 

Looking at the diagram, note that a certain portion of the area of the circle has been cut out to form the sector. In particular, this is the same as the portion of the total angle that has been cut out to form $\theta$θ, and is also the same as the portion of the circumference that is the arc length $l$l.

That is, 

$\frac{\text{Area of sector}}{\text{Area of circle}}=\frac{\theta}{\text{Total angle size}}=\frac{l}{\text{Circumference of circle}}$Area of sectorArea of circle​=θTotal angle size​=lCircumference of circle​.

Now we know that the total angle measure of a circle is $360^\circ$360° and that the circumference of a circle of radius $r$r is $2\pi r$2πr, so we can rewrite the second part of this equality as

$\frac{\theta}{360^\circ}=\frac{l}{2\pi r}$θ360°​=l2πr​.

Rearranging, we can see that

$l=2\pi r\times\frac{\theta}{360^\circ}$l=2πr×θ360°​.

That is, the arc length is equal to the portion $\frac{\theta}{360^\circ}$θ360°​ of the total length of the circumference.

 

 

Practice questions

Question 1

Question 2

Question 3