SOLUTION:
Case: Arc length and Area of sectors
Given: The length of the arc and the angle subtended
Required: To find the area of the sector
Method:
Step 1: First we use the length of the arc to get the radius of the circle
[tex]\begin{gathered} l_{arc}=\text{ }\frac{8\pi}{9},\text{ }\theta=80\degree \\ l_{arc}=\text{ }\frac{\theta}{360}\times2\pi r \\ \frac{8\pi}{9}=\text{ }\frac{80}{360}\times2\pi r \\ \frac{8\pi}{9}=\text{ }\frac{160\pi r}{360} \\ Cross\text{ multiplying} \\ 1440\pi r=2880\pi \\ r=\text{ }\frac{2880\pi}{1440\pi} \\ r\text{ =2} \end{gathered}[/tex]
Step 2: Use the radius to find the area of the sector
[tex]\begin{gathered} A_{sector}=\text{ }\frac{\theta}{360}\pi r^2 \\ A_{sector}=\frac{80}{\text{360}}\pi(2)^2 \\ A_{sector}=\frac{80}{\text{360}}\pi\times4 \\ A_{sector}=\frac{320}{\text{360}}\pi \\ A_{sector}=\frac{8}{9}\pi \end{gathered}[/tex]
Final answer:
The Area of the sector in square units is
[tex]\frac{8\pi}{9}[/tex]
