Question

Please await me in finding the area of the shaded portion of this figure.in terms of pi

Answer 1

Soluion:

Given the figure below:

OAB is congruent to OCD.

Thus, to evaluate the area of the shaded rortion, we have

`2(area\text{ of the sector - area of the triangle OAB\rparen}`

Step 1: Evluate the area of tehe sector OAB.

The area of the sector of a sector is expressed as

`\begin{gathered} Area_(sector)=(\theta)/(360)*\pi r^2 \\ where \\ \theta\Rightarrow central\text{ angle} \\ r\Rightarrow radius\text{ of the circle} \end{gathered}`

Thus,

`\begin{gathered} central\text{ angle AOB= 180-90} \\ =90 \\ radius,\text{ r=6 ft.} \\ thus, \\ Area_(sector)=(90)/(360)*\pi*6\text{ ft}*6\text{ ft} \\ =9\pi\text{ square feet} \end{gathered}`

Step 2: Evaluate the area of the triangle OAB.

The area of the triangle OAB is expressed as

`\begin{gathered} Area_(triangle)=(1)/(2)ab\sin C \\ where \\ a\Rightarrow OA \\ b\Rightarrow OB \\ C\Rightarrow\angle O \\ OA=OB=6\text{ ft} \\ thus, \\ Area_(triangle)=(1)/(2)*6ft*6ft*\sin90 \\ =18\text{ square feet} \end{gathered}`

Step 3: Evaluate the area of the shaded portion.

Recall that OAB is congruent to OCD,

`OAB\cong OCD`

Thus, we have the area of the shaded portion to be evaluated as

Hence, the area of the shaded region is

`18(\pi-2)\text{ square feet}`