Problem 5. Three regular heptagons have a common center, their sides are respectively parallel. The sides of two heptagons are 6 cm and 30 cm. The third heptagon divides the area of the figure enclosed between the first two in the ratio $1: 5$, counting from the smaller heptagon. Find the side of the third heptagon.
Answer: $6 \sqrt{5}$.

Solution: Let the side of the third heptagon be $x$. The areas of similar figures are in the ratio of the squares of corresponding sides. Therefore, the areas enclosed between the heptagons are in the ratio $\left(x^{2}-6^{2}\right):\left(30^{2}-x^{2}\right)=1: 5$, from which the answer follows.