Question

A polygon of nine sides, each of length $2$ is inscribed in a circle. Prove that the radius of the circle is$\csc \left(\frac{\pi }{9}\right)$

Answer 1

Let $AB=2$ unit be one of the sides of the polygon.
Then, $\angle AOB=\frac{2\pi }{9}$
where $O$ is the centre of the circle.
If $OL\perp AB,$ then $AL=1$ and $\angle AOL=\frac{\pi }{9}$
$\therefore$ Radius of the circle$=OA=AL\csc \frac{\pi }{9}$
$=\csc \frac{\pi }{9}$