3) In the square alongside, the arcs are all quarter-circles and, in pairs, have their endpoints in common. The ratio between the perimeter of the gray figure and the perimeter of the square
\(\begin{array}{lll}\text { (A) is } \frac{1}{4} & \text { (B) is } \frac{1}{\pi}\end{array}\)
(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_f693bde306ad8eee0381g-1.jpg?height=264&width=257&top_left_y=1427&top_left_x=1184)
3) 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 $\frac{\pi}{4}$.