### Arc Length and Area of a Sector

#### Definition

Consider a circle with radius $r$, and a portion of its circumference bounded by two radii at an angle of $\theta$ radians apart. The enclosed portion of the boundary is called an arc, and the area enclosed by the arc and the two radii is a sector of the circle.

The length of the arc is

$\qquad \text{Arc length } = r\theta$

The area of the sector is

$\qquad \text{Sector area}= \dfrac{1}{2} r^2 \theta$

Note: $2\pi \text{ radians} = 360^{\circ}$.

#### Worked Examples

###### Example 1

In a circle of radius $4$cm, find the arc length corresponding to the following angles:

a) $\theta = \pi$

b) $\theta = \dfrac{2}{5} \pi$

###### Solution

a) $\theta = \pi$. \begin{align} l &= r\theta \\ &= 4 \times \pi \\ &=12.6 \text{cm (to 3 sig.fig.)} \end{align}

b) $\theta = \dfrac{2}{5} \pi$. \begin{align} l &= r\theta \\ &= 4 \times \dfrac{2}{5} \pi \\ &= 5.02 \text{cm (to 3 sig.fig.)} \end{align}

###### Example 2

Find the length of an arc with angle $\theta = \dfrac{3}{2} \pi$ radians, for the following radii:

a) $r=2$cm.

b) $r=5.5$cm.

###### Solution

a) $r=2$cm. \begin{align} l &= r\theta \\ &= 2 \times \dfrac{3}{2}\pi \\ &= 9.42 \text{cm (to 3 sig.fig.)} \end{align}

b) $r=5.5$cm. \begin{align} l &= r\theta \\ &= 5.5 \times \dfrac{3}{2}\pi \\ &= 25.9 \text{cm (to 3 sig.fig.)} \end{align}

###### Example 3

Given the angle $\theta = \dfrac{4}{5}\pi$ radians and the arc length $l=14$cm, find the radius.

###### Solution

\begin{align} l &=r\theta \\ \\ 14 &= r\times \frac{4}{5}\pi. \\ r &= \frac{14 \times 5}{4 \times \pi} \\ r &= \frac{70}{4 \pi} \\ r &= 5.57 \text{cm (to 3 sig.fig)} \end{align}

###### Example 4

Find the area of the sector of angle $\theta = \dfrac{1}{3} \pi$ radians in a circle of radius $r=6$cm.

###### Solution

\begin{align} A&=\frac{1}{2}r^2 \theta\\ &=\frac{1}{2} \times 6^2 \times \frac{1}{3} \pi\\ &=6\pi\\ &=18.8 \text{cm² (to 3 sig.fig.)} \end{align}

###### Example 5

Given the radius of a circle is $r=3.5$cm and the area of a sector is $A=10$. Find the angle $\theta$.

###### Solution

\begin{align} A& = \frac{1}{2}r^2 \theta\\ 10& = \frac{1}{2} \times 3.5^2 \times \theta\\ 10& = \frac{1}{2} \times 12.25 \times \theta\\ 10& = 6.125 \times \theta\\ \theta &= \frac{10}{6.125}\\ \theta &= 1.63 \text{ radians (to 3 sig.fig.)} \end{align}

#### Video Example

Hayley Bishop finds the arc length and area of a sector of a circle with radius $3$cm, bounded by an angle $\theta = \dfrac{\pi}{5}$.