Solution:
Given:
To evaluate the area of the shaded portion, we subtract the area of the triangle OAB from the area of the sector of the circle.
Thus,
Step 1: Evaluate the area of the sector of the circle.
The area of the sector of a circle is expressed as
[tex]\begin{gathered} Area_{sector}=\frac{\theta}{360}\times\pi r^2 \\ where \\ \theta\Rightarrow central\text{ angle of the sector} \\ r\Rightarrow radius\text{ of the circle} \end{gathered}[/tex]
Thus,
[tex]\begin{gathered} \theta=180-(30+30)=180-60=120 \\ r=6\text{ in.} \\ thus, \\ Area_{sector}=\frac{120}{360}\times\pi\times6\text{ in}\times6\text{ in} \\ =12\pi\text{ square inches} \end{gathered}[/tex]
Step 2: Evaluate the area of the triangle OAB.
The area of the triangle is evaluated as
[tex]\begin{gathered} Area_{triangle}=\frac{1}{2}ab\sin C \\ where \\ a\Rightarrow OA \\ b=OB \\ C\Rightarrow\angle O \end{gathered}[/tex]
Thus, we have
[tex]\begin{gathered} OA=OB=6\text{ in.} \\ \angle O=120 \\ Thus, \\ Area_{triangle}=\frac{1}{2}\times6\text{ in}\times6\text{ in}\times\sin120 \\ =9\sqrt{3}\text{ square inches} \end{gathered}[/tex]
Step 3: Evaluate the area of the shaded region.
Thus, we have
[tex]\begin{gathered} Area\text{ of shaded region = area of sector - area of triangle} \\ =12\pi\text{ -9}\sqrt{3} \\ \end{gathered}[/tex]
Hence, the area of the shaded portion is
[tex]12\pi\text{ -9}\sqrt{3}\text{ square inches}[/tex]
