10.127. A circle is inscribed in an equilateral triangle with a side length of $a$, and a regular hexagon is inscribed in this circle. Find the area of the hexagon.
Solution.

The radius of the circle inscribed in an equilateral triangle with side $a$ is $r=\frac{a \sqrt{3}}{6}$. The side of the hexagon inscribed in this circle is equal to the radius: $a_{6}=r=\frac{a \sqrt{3}}{6}$, and the radius of the circle inscribed in this hexagon, $r=\frac{a_{6}}{2 \operatorname{tg} \frac{180^{\circ}}{6}}=\frac{a_{6} \sqrt{3}}{2}=\frac{a \cdot \sqrt{3} \cdot \sqrt{3}}{6 \cdot 2}=\frac{a}{4}$.

The area of the hexagon $S=\frac{6 \cdot a_{6} \cdot r}{2}=\frac{6 \cdot a \sqrt{3} \cdot a}{2 \cdot 6 \cdot 4}=\frac{a^{2} \sqrt{3}}{8}$.

Answer: $\frac{a^{2} \sqrt{3}}{8}$.