A regular $n$-sided polygon has an area of $t$; what is the area of the annulus formed by the circles inscribed in and circumscribed around the $n$-sided polygon? For example, $n=11, t=107.05 \mathrm{dm}^{2}$.
Let the radius of the circle inscribed in the $n$-sided polygon be $r$, and the radius of the circle circumscribed around the $n$-sided polygon be $R$, then the area of the annulus is:

$$
T=\left(R^{2}-r^{2}\right) \pi
$$

But

$$
R=\frac{r}{\cos \frac{180^{\circ}}{n}}
$$

so

$$
T=\pi r^{2} \frac{1-\cos ^{2} \frac{180^{\circ}}{n}}{\cos ^{2} \frac{180^{\circ}}{n}}=\pi r^{2} \operatorname{tg}^{2} \frac{180^{\circ}}{n}
$$

Furthermore:

$$
t=\frac{n a r}{2}=n r^{2} \operatorname{tg}^{2} \frac{180^{\circ}}{n}
$$

therefore

$$
r^{2}=\frac{t}{n \operatorname{tg} \frac{180^{\circ}}{n}}
$$

Substituting this into (2), we get:

$$
T=\frac{\pi t}{n} \cdot \operatorname{tg} \frac{180^{\circ}}{n}
$$

Substituting the given values, we find that $T=8.9772 \mathrm{dm}^{2}$.

(Kiss Ernö, Budapest, VIII. district.)

The problem was also solved by: Bánó L., Bauer E., Blum I., Cukor G., Ehrenfeld N., Ehrenstein P., Epstein K., Erdélyi I., Esztó P., Fekete M., Fodor H., Földes R., Freund E., Füstös P., Gúman J., Heimlich P., Horti V., Jánosy Gy., Kirchknopf E., Kovács Gy., Krémusz R., Lusztig M., Murarik A., Paunz A., Pető L., Rosenthal M., Sárközy P., Schuster Gy., Sebestyén I., Seligman A., Spitzer L., Szilas O., Travnik L.