Question

In circle F, the length of GH = grand m/GPH = 80°. Find the area shadedbelow. Express your answer as a fraction times T.GH

Answer 1

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

`\begin{gathered} l_(arc)=\text{ }(8\pi)/(9),\text{ }\theta=80\degree \\ l_(arc)=\text{ }(\theta)/(360)*2\pi r \\ (8\pi)/(9)=\text{ }(80)/(360)*2\pi r \\ (8\pi)/(9)=\text{ }(160\pi r)/(360) \\ Cross\text{ multiplying} \\ 1440\pi r=2880\pi \\ r=\text{ }(2880\pi)/(1440\pi) \\ r\text{ =2} \end{gathered}`

Step 2: Use the radius to find the area of the sector

`\begin{gathered} A_(sector)=\text{ }(\theta)/(360)\pi r^2 \\ A_(sector)=\frac{80}{\text{360}}\pi(2)^2 \\ A_(sector)=\frac{80}{\text{360}}\pi*4 \\ A_(sector)=\frac{320}{\text{360}}\pi \\ A_(sector)=(8)/(9)\pi \end{gathered}`

Final answer:

The Area of the sector in square units is

`(8\pi)/(9)`