10.130. The side of a square inscribed in a circle cuts off a segment, the area of which is equal to $(2 \pi-4) \mathrm{cm}^{2}$. Find the area of the square.
Solution.

Let $R$ be the radius of the circle: Then the area of the sector is $\frac{\pi R^{2}}{4}$, and

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-617.jpg?height=446&width=532&top_left_y=77&top_left_x=110)

Fig. 10.118

the area of the triangle is $\frac{R^{2}}{2}$, and therefore, the area of the segment is $\frac{\pi R^{2}}{4}-\frac{R^{2}}{2}=\frac{R^{2}(\pi-2)}{4}$. We have $\frac{R^{2}(\pi-2)}{4}=2 \pi-4$, from which $R=2 \sqrt{2}$ (cm). Thus, the area of the square is $2 R^{2}=16\left(\mathrm{~cm}^{2}\right)$.

Answer: $\quad 16 \mathrm{~cm}^{2}$.