10.152. A circle is inscribed in a regular hexagon with side length $a$, and another circle is circumscribed around the same hexagon. Determine the area of the circular ring enclosed between these two circles.
## Solution.

The radius of the inscribed circle in the hexagon $r_{1}=\frac{a}{2 \operatorname{tg} \frac{180^{\circ}}{6}}=\frac{a \sqrt{3}}{2}$.

The radius of the circumscribed circle around the hexagon $r_{2}=a$. The area of the larger circle $S_{2}=\pi a^{2}$, the smaller one $-S_{1}=\pi \frac{3 a^{2}}{4}$.

The area of the ring $S=S_{2}-S_{1}=\pi a^{2}-\frac{3 \pi a^{2}}{4}=\frac{\pi a^{2}}{4}$.

Answer: $\frac{\pi a^{2}}{4}$.