10.159. Find the ratio of the areas of an equilateral triangle, a square, and a regular hexagon, the lengths of whose sides are equal.
## Solution.

Let the side of the equilateral triangle, square, and hexagon be $a$. Then $S_{3}=\frac{a^{2} \sqrt{3}}{4}, S_{4}=a^{2}, S_{6}=\frac{6 \sqrt{3} a^{2}}{4}$. Therefore,

$$
S_{3}: S_{4}: S_{6}=\frac{a^{2} \sqrt{3}}{4}: a^{2}: \frac{6 \sqrt{3}}{4} a^{2}=\sqrt{3}: 4: 6 \sqrt{3}
$$

Answer: $\sqrt{3}: 4: 6 \sqrt{3}$.