Problem 5. Maxim laid out a polygon on the table using 9 squares and 19 equilateral triangles (without overlapping them) from a set where the sides of all squares and triangles are 1 cm. Could the perimeter of this polygon be 15 cm? [9 points] (M.A. Volchkevich)
Answer. Yes, it could (see fig.).
![](https://cdn.mathpix.com/cropped/2024_05_06_b15c0281e21ba8d77df7g-4.jpg?height=342&width=748&top_left_y=224&top_left_x=352)

Comments. 1. The difficulty here is that the figure is proposed to be constructed with a rather small perimeter: even if we were to construct a polygon only from squares, the perimeter would be no less than 12 cm, and we still need to add 19 whole triangles.

Of all the figures with the given area, the one with the smallest perimeter is a circle. Therefore, if such a polygon exists, it should probably be close to a circle.

By the way, we can calculate that the perimeter of a circle equal in area to our polygon is approximately 14.7 cm. So, it is impossible to obtain a polygon with an even smaller perimeter.