Question

Find the area of the shaded region given in the figure:

Answer 1

Solve for area of the given shaded region:

Given, side of $\square$$ABCDs=14cm$

From the figure, we can say that,

Area of shaded region $=$ Area of $\square$ $ABCD$$-$( Area of four semi-circles $+$ Area of $\square$$EFGH$)

Step 1: Find area of $\square$$ABCD$

Area of $\square$ $ABCD$$={\left(s\right)}^2$

$={\left(14\right)}^2$

$\Rightarrow$Area of $\square$$ABCD$$=196c{m}^2$

Step 2: Find area of semi-circle

From the figure, we can say, diameter of each semi-circle $=$ side of $\square$$EFGH$ and

$AB=3cm+2\times$radius of semi-circle $+$ side of $\square$$EFGH+3cm$

$\Rightarrow 14=3+2r+2r+3$ [ where, $r$ is the radius of each semi-circle]

$$\begin{gathered} \Rightarrow 4r=8 \\ \Rightarrow r=2cm \end{gathered}$$

$\therefore$ Area of each semi-circle$\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathit{\pi }{\mathit{r}}^2$

$=\frac{1}{2}\times \frac{22}{7}\times 4=\frac{44}{7}c{m}^2$

$\Rightarrow$Area of four semi-circles$=4\times \frac{44}{7}=25.14c{m}^2$

Step 3: Find area of $\square$ $EFGH$

Side of $\square$ $EFGH$$=2r=2\left(2\right)=4cm$

$\therefore$Area of $\square$ $EFGH$$={\left(4\right)}^2=16c{m}^2$

Step 4: Find area of shaded region

Area of shaded region $=$ Area of $\square$ $ABCD$$-$( Area of four semi circles $+$ Area of $\square$$EFGH$)

$\Rightarrow$Area of shaded region $=196-\left(25.14+16\right)=154.86c{m}^2$

Hence, area of the shaded region is $154.86c{m}^2$