1.55. Inside a square with side $a$, a semicircle is constructed on each side as a diameter. Find the area of the rosette bounded by the arcs of the semicircles.
1.55. Chord $O A$ subtends an arc of $90^{\circ}$ (Fig. 1.43); therefore, the area of half a petal is $\frac{\pi a^{2}}{16}-\frac{a^{2}}{8}=\frac{a^{2}(\pi-2)}{16}$. Consequently, the desired area is

$$
S=8 \cdot \frac{a^{2}(\pi-2)}{16}=\frac{a^{2}(\pi-2)}{2}
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-438.jpg?height=373&width=373&top_left_y=1411&top_left_x=781)

Fig. 1.43

Answer: $0.5 a^{2}(\pi-2)$.