To answer this question we need:
1. Find the area of the square.
2. Find the diameter of the circle. This is the diagonal of the square.
3. Find the half of the diameter, and then solve for the area of the circle.
4. Subtract the area of the square from the area of the circle.
Then, we have:
Area of the square
The area of the square is:
![A_{\text{square}}=s^2=(6\operatorname{cm})^2=36\operatorname{cm}^2]()
Diameter of the circle (diagonal of the square)
We can use the Pythagorean Theorem. We have a right triangle.
![d^2=(6\operatorname{cm})^2+(6\operatorname{cm})^2=36\operatorname{cm}+36\operatorname{cm}=72\operatorname{cm}^2]()
Extracting the square root to both sides of the equation:
![\sqrt[]{d^2}=\sqrt[]{72\operatorname{cm}^2}\Rightarrow d=\sqrt[]{2^2\cdot2\cdot3^2}=2\cdot3\cdot\sqrt[]{2}=6\cdot\sqrt[]{2}\Rightarrow d=6\cdot\sqrt[]{2}]()
The radius of the circle is, therefore:
![r=(d)/(2)=\frac{6\cdot\sqrt[]{2}}{2}=3\cdot\sqrt[]{2}](https://img.qammunity.org/qa-images/2023/formulas/mathematics/college/ab9ewn4xxb8g8rcm6nya.png)
Area of the circle
It is given by:
![A_{\text{circle}}=\pi\cdot r^2=\pi\cdot(3\cdot\sqrt[]{2})^2=\pi\cdot3^2\cdot2=\pi\cdot18=18\pi\approx56.5486677646](https://img.qammunity.org/qa-images/2023/formulas/mathematics/college/h8cgfndo8yprhchn5a39.png)
Hence, the shaded area is, then:
![A_{\text{shadedarea}}=A_{\text{circle}}-A_{\text{square}}=56.5486677646\operatorname{cm}-36\operatorname{cm}^2=20.5486677646]()
In summary, the shaded area is equal to, approximately, 20.5 sq. centimeters.