General formula for the upper bound of pi involving nested square roots (circumscribed perimeters of regular polygons) The formula for the lower bound of pi involving nested square roots looks like this: $p_{2^m} = 2^m\sqrt{2-\sqrt{2+\sqrt{2+ \sqrt{2+...}}}}$ where there are $m-1$ nested square roots.
For example, inscribing a square in a unit circle has $m = 2$, so the perimeter $p_4 = 2^2\sqrt2 = 4\sqrt2 < 2\pi$. Inscribing an octagon would have $m = 3$, so $p_8 = 2^3\sqrt{2-\sqrt2}$.
Deriving this formula, which seems to have significant relation to Viete's formula, involves inscribing a figure inside a circle and using various theorems related to circles and chords. I will upload the page to give some context.
I have been attempting to derive a formula for the upper bound, involving circumscribing figures about a unit circle. This seems to present more challenge to me, and I would greatly appreciate a step in the right direction. Please no trigonometry or angles, as the expression for pi must only involve these nested square roots.
I think most of my difficulty lies in sketching a decent figure, as it is not that geometrically obvious what a circumscribed $s_8$ would look like given a circumscribed $s_4$.

 A: If $\theta$ is the half-angle subtened by each side of the polygon, you have
$\cos\theta=(1/2)\sqrt{2+\sqrt{2+\sqrt{2+...}}}.$
For comparison the factor in your lower-bound expression is
$\sin\theta=(1/2)\sqrt{2\color{blue}{-}\sqrt{2+\sqrt{2+...}}},$
which differs only by the sign indicated in blue. You should see that the squares of the sine and cosine properly add up to $1$.
The factor you want in the upper bound expression is $\tan\theta$, which is the second of these expressions divided by the first.
So, take your lower bound, multiply in one more power of $2$, and divide by the appropriate iteration of $\sqrt{2+\sqrt{2+\sqrt{2+...}}}$ (same length as the radical in the lower bound expression). Thus for the octagon you have the lower bound
$\pi_{-,8}=2^2\sqrt{2-\sqrt2}\approx3.061,$
and so the corresponding upper bound would be
$\pi_{+,8}=\dfrac{2^3\sqrt{2-\sqrt2}}{\sqrt{2+\sqrt2}}\approx3.314.$
A: I hope this paper will provide an answer to your question: https://doi.org/10.1007/s11139-018-9996-8
Here is a preprint version:
https://doi.org/10.48550/arXiv.1610.07713
