Soluion:
Given the figure below:
OAB is congruent to OCD.
Thus, to evaluate the area of the shaded rortion, we have
[tex]2(area\text{ of the sector - area of the triangle OAB\rparen}[/tex]
Step 1: Evluate the area of tehe sector OAB.
The area of the sector of a sector is expressed as
[tex]\begin{gathered} Area_{sector}=\frac{\theta}{360}\times\pi r^2 \\ where \\ \theta\Rightarrow central\text{ angle} \\ r\Rightarrow radius\text{ of the circle} \end{gathered}[/tex]
Thus,
[tex]\begin{gathered} central\text{ angle AOB= 180-90} \\ =90 \\ radius,\text{ r=6 ft.} \\ thus, \\ Area_{sector}=\frac{90}{360}\times\pi\times6\text{ ft}\times6\text{ ft} \\ =9\pi\text{ square feet} \end{gathered}[/tex]
Step 2: Evaluate the area of the triangle OAB.
The area of the triangle OAB is expressed as
[tex]\begin{gathered} Area_{triangle}=\frac{1}{2}ab\sin C \\ where \\ a\Rightarrow OA \\ b\Rightarrow OB \\ C\Rightarrow\angle O \\ OA=OB=6\text{ ft} \\ thus, \\ Area_{triangle}=\frac{1}{2}\times6ft\times6ft\times\sin90 \\ =18\text{ square feet} \end{gathered}[/tex]
Step 3: Evaluate the area of the shaded portion.
Recall that OAB is congruent to OCD,
[tex]OAB\cong OCD[/tex]
Thus, we have the area of the shaded portion to be evaluated as
[tex]\begin{gathered} 2(area\text{ of the sector - area of the tr}\imaginaryI\text{angle OAB}\operatorname{\rparen} \\ =2(9\pi-18) \\ =18(\pi-2)\text{ square feet} \end{gathered}[/tex]
Hence, the area of the shaded region is
[tex]18(\pi-2)\text{ square feet}[/tex]
