7. A circle is circumscribed around a rectangle. The lengths of the adjacent sides of the rectangle are in the ratio 4 : 3. The area of the rectangle is $108 \mathrm{~cm}^{2}$. Calculate the area of the unshaded part of the circle.

![](https://cdn.mathpix.com/cropped/2024_06_03_00a3c3ecfcf7f36b93a9g-2.jpg?height=496&width=505&top_left_y=1037&top_left_x=570)

The use of a pocket calculator or any reference materials is not allowed.
7. 

![](https://cdn.mathpix.com/cropped/2024_05_30_32ee0b570cef3700b3ccg-2.jpg?height=486&width=488&top_left_y=1373&top_left_x=287)

From the conditions of the problem, we have $a \cdot b = 108$ and $a : b = 4 : 3$.

From the second equation, it follows that $a = \frac{4}{3} b$,

so $\frac{4}{3} b \cdot b = 108$ or $b = 9 \text{ cm}$

and $a = 12 \text{ cm}$.

$1 \text{ POINT}$

Using the Pythagorean theorem, we get $9^2 + 12^2 = d^2$ or $d = 15 \text{ cm}$.

Then the radius of the circle circumscribed around the rectangle is $7.5 \text{ cm}$.

$1 \text{ POINT}$

The area of the unshaded part is equal to the area of the circle minus the area of the shaded rectangle, i.e., $P = (7.5)^2 \pi - 108$.