A polygon has a perimeter of $800 \mathrm{~cm}$. What is the length of one side of a square whose area is equal to the area of the polygon, if each side of the polygon is tangent to a circle with a radius of $100 \mathrm{~cm}$?
For a polygon with perimeter $K$ and an inscribed circle with radius $r$, the area of the polygon is

$$
T=\frac{K \cdot r}{2}
$$

In this case,

$$
T=4000 \mathrm{~cm}^{2}
$$

thus the side of the sought square is $200 \mathrm{~cm}$.

(Neumann Frida, Budapest.)

The problem was also solved by: Bauer E., Czigler I., Dénes M., Erdős V., Frankl F., Fried E., Grünvald Gy., Koffler B., Köhler I., Neumann L., Rosenberg E., Sichermann F., Spitzer L., Szántó L., Szilárd V., Vilcsek A., Viola R., Wellis E.