10.030. On the sides of a square, equilateral triangles are constructed outside it, and their vertices are sequentially connected. Determine the ratio of the perimeter of the resulting quadrilateral to the perimeter of the given square.
Solution.

Let the side of the square be $x$, then the perimeter of the square

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-570.jpg?height=664&width=1154&top_left_y=94&top_left_x=93)

$P_{0}=4 x, \triangle F B M$ is isosceles, $\angle F B M=360^{\circ}-90^{\circ}-2 \cdot 60^{\circ}=150^{\circ}$, $F M=\sqrt{2 x^{2}-2 x^{2} \cdot \cos 150^{\circ}}=x \sqrt{2+\sqrt{3}}$ (Fig. 10.30). Thus,

$$
P_{E F M N}=4 x \sqrt{2+\sqrt{3}}, \frac{P}{P_{0}}=\sqrt{2+\sqrt{3}}=\frac{\sqrt{6}+\sqrt{2}}{2}
$$

Answer: $\frac{\sqrt{6}+\sqrt{2}}{2}$.