Given a regular $n$-gon with sides of unit length, what is the side length of the smallest regular $n+1$-gon containing it?
For $n=3$ a bit of calculus yields a square of side length $$\cos\frac{\pi}{24}=\frac{1+\sqrt{3}}{2\sqrt{2}}\approx0.965925826289...$$ For $n=4$ a bit more calculus and a few more cases to check yield a regular pentagon of side length $$\frac{\sin\tfrac{7\pi}{40}}{\sin\tfrac{3\pi}{10}}=\frac{\sqrt{5-\sqrt{5}+\sqrt{25-10\sqrt{5}}}}{\sqrt{5}}\approx0.936859701734...$$ What is the minimal side length for $n=5$? I hope someone, somewhere has already taken the time to publish a list values for small $n$. Any reference is welcome, though I'll be satisfied with any effective method to compute the exact values as well.
If a regular n-gon is inscribed in a circle of radius $r$,, the central angle of a side is $2\pi/n$, so the distance of the side from the center $r_i(n)$ satisfies $r_i(n) = r\cos(\pi/n) $ and the length of the side is $s_i =2r\sin(\pi/n) $.
If a regular n-gon is circumscribed around a circle or radius $r$, the central angle of a side is $2\pi/n$, so the distance of the corner from the center $s(n)$ satisfies $r_c(n) = r/\cos(\pi/n) $ and the length of the side is $s_c =2r\tan(\pi/n) $.
If the sides of a regular n-gon are of unit length, the distance to the vertices $d(n)$ satisfies $\dfrac{1/2}{d(n)} =\sin(\pi/n) $ so $d(n) =\dfrac1{2\sin(\pi/n)} $.
If this is the distance to the side of a regular m-gon (I will set $m = n+1$ later), then, if $s$ is the side of the m-gon then $\dfrac{s/2}{d(n)} =\tan(\pi/m) $ so $s =2d(n)\tan(\pi/m) =2\dfrac1{2\sin(\pi/n)}\tan(\pi/m) =\dfrac{\tan(\pi/m)}{\sin(\pi/n)} $.
If $m = n+1$, this is $\dfrac{\tan(\pi/(n+1))}{\sin(\pi/n)} $.
This, of course, does not take into account the possibility that the outer m-gon might be able to be smaller if it is rotated, so this is an upper bound.
If we use the approximation $\sin(x) \approx x$, this is about $\dfrac{n}{m} $.
If the side of the m-gon is the same of the side of the n-gon, then $\tan(\pi/m) =\sin(\pi/n) $ so $\dfrac{\pi}{m} =\arctan(\sin(\pi/n)) $ or $m =\dfrac{\pi}{\arctan(\sin(\pi/n))} $.