37. On the sides of a square, semicircles are constructed with the sides as diameters (Fig. 11). Determine the area of the shaded figure (four petals), if the side of the square is $a$.
37. The area of a semicircle is $\frac{\pi a^{2}}{8}$, and the sum of the areas of four such semicircles is $\frac{\pi a^{2}}{2}$. The area of the square is $a^{2}$. Therefore, the desired area is:

$$
\frac{\pi a^{2}}{2}-a^{2} \approx a^{2}\left(\frac{11}{7}-1\right)=\frac{4}{7} a^{2} . \quad\left(\frac{\pi}{2} \approx \frac{11}{7}\right)
$$

182
![](https://cdn.mathpix.com/cropped/2024_05_21_00604dc020e3721fc1f4g-184.jpg?height=858&width=592&top_left_y=227&top_left_x=318)

Fig. 31
![](https://cdn.mathpix.com/cropped/2024_05_21_00604dc020e3721fc1f4g-184.jpg?height=844&width=810&top_left_y=228&top_left_x=925)

Fig. 32

![](https://cdn.mathpix.com/cropped/2024_05_21_00604dc020e3721fc1f4g-184.jpg?height=837&width=837&top_left_y=1215&top_left_x=564)

Fig. 33