The Apothem $r$ and Radius $R$ of a regular polygon with $n$ sides and Side length $s$ are
$$r = \frac{s}{2} \cdot \cot \left(\frac{\pi}{n}\right)$$
$$R = \frac{s}{2} \cdot \csc \left(\frac{\pi}{n}\right)$$
The Annulus area $A$ between the Circumcircle and Incircle is
\begin{align*} A &= \pi R^2 - \pi r^2 \\ &= \pi (R^2 - r^2) \\ &= \pi \left( \left(\frac{s}{2} \cdot \csc \left(\frac{\pi}{n}\right)\right)^2 - \left(\frac{s}{2} \cdot \cot \left(\frac{\pi}{n}\right)\right)^2 \right) \\ &= \frac{\pi s^2}{4} \left( \csc^2\left(\frac{\pi}{n}\right) - \cot^2\left(\frac{\pi}{n}\right) \right) \\ &= \frac{\pi s^2}{4} \end{align*}
Rearranging, $\pi$ can be defined as
$$\pi = \frac{{4A}}{{s^2}}$$
Question
Is that correct? It seems odd that the area is related to the side $s$ and not $n$. Also, $\pi$ can be expressed in terms of the area and sides, unless there's some circular reasoning?