Question

In the given figure, $ ABCD$ is a square of side $ 14 cm$. With centres $ A$, $ B$, $ C$ and $ D$, four circles are drawn such that each circle touch externally two of the remaining three circles. Find the area of the shaded region.

Answer 1

Step 1: Find area of the quarter circle

The radius of the quarter circle is $\raisebox{1ex}{$AB$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

Therefore, the area of the quarter circle $=\frac{1}{4}\times \mathrm{\pi }\times {\mathrm{r}}^2\left[\mathrm{r}=\mathrm{radius}\right]$

$$\begin{gathered} =\frac{1}{4}\times \frac{22}{7}\times {\left(\frac{AB}{2}\right)}^2 \\ =\frac{1}{4}\times \frac{22}{7}\times {\left(\frac{14}{2}\right)}^2 \\ =38.5c{m}^2 \end{gathered}$$

Step 2. Find area of the square

The length of each side of the square is $14cm$

The area of the square is $=l^2\left[l=\mathrm{length}\mathrm{of}\mathrm{each}\mathrm{side}\mathrm{of}\mathrm{square}\right]$

$$\begin{gathered} ={14}^2 \\ =196c{m}^2 \end{gathered}$$

Step 3. Find area of the shaded region

$$\begin{gathered} \mathrm{The}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{shaded}\mathrm{region}=\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{square}-4\times \mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{quarter}\mathrm{circle} \\ \Rightarrow \mathrm{The}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{shaded}\mathrm{region}=196-4\times \left(38.5\right) \\ \Rightarrow \mathrm{The}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{shaded}\mathrm{region}=196-154 \\ \Rightarrow \mathrm{The}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{shaded}\mathrm{region}=42{\mathrm{cm}}^2 \end{gathered}$$

Hence, the area of the shaded region is $42c{m}^2$.