How to define $\pi$ from geometry to the limit of a sequence?

Question

There are several definitions of $\pi$ based on the limit of some sequences or series. (Maybe) The most famous example is the solution of the Basel problem:

$$\pi = \sqrt{6 \sum_{n=1}^{+\infty}\frac{1}{n^2}}.$$

Anyway, even though I've found several of these sequences, none of them seems to be derived by using a constructive geometric approach.

What I mean? For instance, let's consider a unit circle and the regular polyhedra inscribed with $m>2$ sides. Let $s_m$ the side length of the $m$-th polyhedron. Then, we can define $\pi$ as follows:

$$\pi = \frac{1}{2}\lim_{m \to +\infty} m\cdot s_m.$$

I've tried to find out the expression of $s_m$, but I need to define it by using $\pi$ itself, i.e.

$$s_m = \frac{2}{\sin\left(\frac{2\pi}{m}\right)}.$$

So it seems that, in order to evaluate $\pi$, one needs to use $\pi$ itself!!!

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Summing up, is there any sequence/series based on simple geometric arguments that can be used to define $\pi$?

Answer 1

Hint: try to find a recursive relation between $s_{2^{n+1}}$ and $s_{2^n}$

Answer 2

$\Pi_{k=0}^3(i+k)$ where $i^2=-1$ arguably does it in disguise.

This is $i(i+1)(i+2)(i+3)=(-1+i)(5+5i)=-10=10e^{i\pi/2}$

$(i+k)=\sqrt{k^2+1}e^{i\arctan(1/k)}$

You can think of $i$ as a vertical line of unit length starting at the origin. Multiplication by the subsequent terms extends the length of the line and rotates it a bit.

This all implies $\arctan{1/0}+\arctan{1}+\arctan{2}+\arctan{3}$ is a multiple of $\pi/2$. So the some of terms with a geometric meaning not explicitly mentioning $\pi$ add it to a rational multiple of $\pi$.