Question

Find the Arc length and the area of the sector. Leave your answer in reduced fraction form in terms of pi.

Answer 1

Solution

Given

$\begin{gathered} r=9cm \\ _\theta=45^0 \end{gathered}$

Part A

The formula of length of an arc

$Length\text{ of an arc=}\frac{\theta}{360}\times2\pi r$

Substitute the given into the formula

$\begin{gathered} Length\text{ of the arc=}\frac{45}{360}\times2\times\pi\times9 \\ Length\text{ofthearc=}\frac{1}{8}\text{\times2\pi\times9} \\ \\ Length\text{ofthearc=}\frac{1}{8}\times18\pi \\ \\ The\text{ length of the arc=}\frac{9}{2}\pi\text{ or 2}\frac{1}{4}\pi cm \end{gathered}$

Part b

Area of a sector

$\begin{gathered} Area\text{ of sector =}\frac{\theta}{360}\times\pi r^2 \\ \\ \end{gathered}$

Substitute the given into the formula

$\begin{gathered} Area\text{ of sector=}\frac{45}{360}\times\pi\times9^2 \\ \\ Area\text{ofsector=}\frac{1}{8}\times\pi\times81=\frac{81\pi}{8}\text{ or 10}\frac{1}{8}\pi cm^2 \\ \end{gathered}$

Part C

$m\angle AB=45^0$