Given a regular 15-sided polygon with side length $2a$. What is the area of the annulus (ring-shaped region) bounded by the inscribed and circumscribed circles of the 15-sided polygon?
Let the common center of the circumscribed and inscribed circles of the 15-sided polygon be $O$, their radii $R$ and $r$ respectively. Thus, the area of the annulus $t = R^{2} \pi - r^{2} \pi = \pi (R^{2} - r^{2})$. Let one side of the polygon be $AB = 2a$, and its midpoint be $F$. Thus, from the right triangle $OAF$, we have $R^{2} - r^{2} = a^{2}$, which is known, so $t = \pi a^{2}$.

![](https://cdn.mathpix.com/cropped/2024_05_02_94873e252b687fefb0e0g-1.jpg?height=379&width=426&top_left_y=304&top_left_x=833)

In our reasoning, we only used the fact that the side of the polygon is a chord of the outer boundary circle of the annulus that touches the inner one. Thus, we obtained this: the area of the annulus is equal to that of a circle whose diameter is as large as the largest chord of the outer boundary circle that lies within the annulus.

Kersner Róbert (Ajka, Bródy I. g. II. o. t.)