10.128. A circle is circumscribed around a square with side length $a$, and a regular hexagon is circumscribed around the circle. Determine the area of the hexagon.
## Solution.

The radius of the circle circumscribed around the square is equal to half the diagonal of the square $r=\frac{a \sqrt{2}}{2}$. This same circle is inscribed in the hexagon: $r=\frac{a_{6}}{2 \operatorname{tg} \frac{180^{\circ}}{6}}=\frac{a_{6} \sqrt{3}}{2}$. We get $\frac{a \sqrt{2}}{2}=\frac{a_{6} \sqrt{3}}{2}$, from which
$a \sqrt{2}$ $a_{6}=\frac{a \sqrt{2}}{\sqrt{3}}$. Then the area of the hexagon

$$
S=\frac{6 \cdot a_{6} \cdot r}{2}=\frac{6 a \sqrt{2} \cdot a \sqrt{2}}{2 \cdot \sqrt{3} \cdot 2}=\sqrt{3} a^{2}
$$

Answer: $\sqrt{3} a^{2}$.