Answer
The area of the shaded region of the given figure, in terms of \[\pi \] is\[\left( \text{36}-9\pi \right)in{{.}^{2}}\]
Work Step by Step
We have to find the area of the shaded region in the given figure. In the figure, four circles are inscribed in a square with side equal to \[6\text{ inches}\]. Therefore, the side of the square equals twice the diameter of a circle, as shown in the figure.
\[\begin{align}
& 2d=6\text{ inches} \\
& \text{ }d\text{ = }3\text{ inches } \\
\end{align}\]
So, the radius will be half of the diameter, that is \[\text{1}\text{.5 inches}\]. The area of shaded region will be found out by subtracting the area of four circles form the square.
According to this formula, the area of one circle, with radius equal to \[1.5\text{ inches}\] will be:
\[\begin{align}
& {{A}_{0}}=\pi {{r}^{2}} \\
& =\pi \times {{\left( \frac{3}{2}\text{ in}\text{.} \right)}^{2}} \\
& \text{=}\frac{9}{4}\pi \text{ in}{{.}^{2}}
\end{align}\]
As there are \[4\]circles with the same radius, the total area of all the circles will be as follows:
\[\begin{align}
& {{A}_{1}}=4\times \frac{9}{4}\pi \\
& =9\pi \text{ in}{{.}^{2}}
\end{align}\]
The area of the square, with side equal to \[6\text{ inches}\]will be:
\[\begin{align}
& {{A}_{2}}=6\text{ in}\times 6\text{ in} \\
& =36\text{ in}{{\text{.}}^{2}}
\end{align}\]
The area of the shaded region will be:
\[\begin{align}
& A={{A}_{2}}-{{A}_{1}} \\
& =\text{36 in}{{.}^{2}}-9\pi \text{ in}{{.}^{2}} \\
& =\left( \text{36}-9\pi \right)in{{.}^{2}}\text{ }
\end{align}\]
Hence, the area of the shaded region of the given figure, in terms of \[\pi \] is\[\left( \text{36}-9\pi \right)in{{.}^{2}}\].
