3) A regular hexagon with side length $2 \mathrm{~m}$ is given. Calculate the area of the circular ring delimited by the inscribed circle and the circumscribed circle of the hexagon.
(A) $\frac{\pi}{2} \mathrm{~m}^{2}$
(B) $\pi \mathrm{~m}^{2}$
(C) $\frac{4 \pi}{3} \mathrm{~m}^{2}$
(D) $2 \pi \mathrm{~m}^{2}$
(E) $\frac{\pi}{9} \mathrm{~m}^{2}$
3. The answer is $\mathbf{( B )}$.

It is necessary to calculate the radius of the inscribed circle and the radius of the circumscribed circle of the hexagon. The radius $R$ of the circumscribed circle is equal to the side of the hexagon, which measures $2 \mathrm{~m}$. The radius $r$ of the inscribed circle is given by the height of an equilateral triangle with a side length of $2 \mathrm{~m}$ (see figure) and thus measures $\sqrt{3} \mathrm{~m}$. Therefore, the area of the circumscribed circle is $4 \pi \mathrm{m}^{2}$ and the area of the inscribed circle is $3 \pi \mathrm{m}^{2}$, and the difference between these, which is the required area, is $\pi \mathrm{m}^{2}$.

[Problem proposed by U. Bindini]

![](https://cdn.mathpix.com/cropped/2024_04_17_ed8bf509241053ca04f5g-1.jpg?height=504&width=507&top_left_y=1952&top_left_x=1449)