I want to find the general formula to calculate the slope of the bold line in the picture when we put one side of the regular $n$-gon upside. I have found $3$ data below, even though I haven't verified the third data. Note that their sides are identical in length. I suppose, there's something connected with tangent. I.e.:
Let $m$ be the slope. I have found that
$$m_{3} = \tan\left(\frac{\pi}{2}\left(\frac13\right)\right)= \frac{\sqrt3}{3}$$
$$m_{4} =\tan\left(\frac{\pi}{2}\left(\frac12\right)\right)= 1$$
$$m_{5} =\tan\left(\frac{\pi}{2}\left(\frac23\right)\right)= \sqrt3$$
And note that:
$$\frac13 < \frac12 < \frac23 < \cdots < 1$$
I give the supremum $1$ because we know $\tan{90^°}$ is undefined. The reason behind it, I believe, because the slope is getting bigger and bigger and we also know that straight vertical line has infinity slope if I understand correctly. Only if I could find the pattern, would I be able to find the formula.
The angle is $\dfrac\pi2-\dfrac12\cdot\dfrac{2\pi}n$. The slope is $$ \tan\big(\dfrac\pi2-\frac12\cdot\dfrac{2\pi}n\big)=\cot\big(\dfrac{ \pi}n\big) $$ For large $n$ (small argument): $$ \cot\big(\dfrac{ \pi}n\big)\to\frac n\pi-\frac\pi{3n}-\frac{\pi^3}{45n^3}+... $$