Answer:
[tex]144\pi\ un^2.[/tex]
Step-by-step explanation:
In the attached diagram, circle R is shown. Line segments QR and SR are radii and QR = SR = 18 units.
The measure of the central angle QRS is [tex]\dfrac{8\pi}{9}[/tex]
1. Find the area of the whole circle:
[tex]A_{circle}=\pi r^2=\pi \cdot 18^2=324\pi \ un^2.[/tex]
2. Note that the whole circle is determined by the full rotation angle with measure [tex]2\pi[/tex] radians. So,
[tex]\begin{array}{cc}\text{Angle}&\text{Area}\\ \\2\pi &324\pi \\ \\\dfrac{8\pi }{9}&A_{sector}\end{array}[/tex]
So, write a proportion:
[tex]\dfrac{2\pi}{\frac{8\pi}{9}}=\dfrac{324\pi}{A_{sector}}[/tex]
Cross multiply
[tex]2\pi \cdot A_{sector}=324\pi \cdot \dfrac{8\pi }{9}\\ \\A_{sector}=162\pi \cdot \dfrac{8}{9}=144\pi\ un^2.[/tex]
