4) In the square alongside, the arcs are all quarter-circles and have, in pairs, common endpoints. The ratio between the perimeter of the gray figure and the perimeter of the square
(A) is $\frac{1}{4}$
(B) is $\frac{1}{\pi}$
(C) is $\frac{\pi}{4}$
(D) is $\frac{1}{2}$
(E) cannot be determined with the given information.

![](https://cdn.mathpix.com/cropped/2024_04_17_807214d42c092eaa073dg-1.jpg?height=269&width=272&top_left_y=145&top_left_x=2497)
4) The answer is (C). Indicating with $l$ the side of the square and with $r$ the radius of the small quarter-circles, the radius of the large ones will be $l-r$. The perimeter of the gray figure will therefore be

$$
2 \times \frac{2 \pi r}{4}+2 \times \frac{2 \pi(l-r))}{4}=\pi r+\pi l-\pi r=\pi l
$$

The perimeter of the square is $4 l$ and the ratio of the perimeters is therefore $\frac{\pi}{4}$.