Arc length

In a nutshell

An arc length is a section of the circumference of a circle. Now that the angle subtending the arc is measured in radians, the formula to calculate the arc length is simple and very easy to use.

Arc length formula

It has been previously stated that an angle $\theta$ measured in radians is given by $\theta = \dfrac {l}{r}$ , where $l$ is the arc length and $r$ is the radius. Rearranging this formula for $l$ gives you the formula for arc length, which is useful if the angle is given in radians.

\boxed{l = r \theta}

$l$ is the arc length.
$r$ is the radius of the circle.
$\theta$ is the angle measured in radians.

Example 1

The shape given has a radius of $8 \ cm$ . Calculate the perimeter of the shape.

First find the angle in the major sector in radians. This can be found by converting from degrees:

$\theta = 270 \degree = \dfrac {3 \pi}{2} \ rads$

Alternatively this is calculated by

$2\pi-\frac{\pi}2=\frac{3\pi}2$

Find the arc length using the formula:

$l = r \theta = 8 \times \dfrac{3 \pi}{2} = 12 \pi$

Find the perimeter:

$P = 12 \pi + 8 + 8 = \underline{53.7 \ cm \ (3 \ s.f.)}$

Example 2

In the sector shown, the perimeter is five times bigger that the arc length. Find the angle $\angle AOB$ .

Write a formula for the perimeter which consists of two radii and the arc length:

$P = r + r + r\theta$ 

The perimeter is five times bigger than the arc length:

$5 \times r \theta = r + r + r\theta$ 

Solve for $\theta$ :

$\begin{aligned}5r \theta &= r + r + r\theta \\4r \theta &= 2r \\\theta &= \dfrac {2r}{4r} \\\theta &= \underline{0.5 \ rads}\end{aligned}$ 