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(this question is related to For what $n$ does a hyperbolic regular $n$-gon exist around a circle? )

Given two concentric circles $ C_r$ with radius $r$ and $C_R$ with radius $R$

what is the lowest regular n -gon (lowest n) that fits between them ? (if i knew that i could answer the related question)

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Willemien
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    Assuming that you want the $n$-gon to touch the inner circle and be inside the bigger one: The ratio of the distances from the center of a regular $n$-gon to a vertex vs. to the midpoint of a side is $1/\cos(\pi/n)$. So it looks like you need $n$ large enough that $R\cos(\pi/n)>r$. – Jyrki Lahtonen May 09 '15 at 05:41

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Here, $r$ is the radius of the inner circle and $R$ is the radius of the outer circle, so $r < R$.

For a circumscribed $n$-gon for a circle of radius $r$, the central half-angle is $\pi/n$, so the distance to the outer point $d$ satisfies $r/d =\cos(pi/n) $ or $d =r/\cos(pi/n) $. Therefore $R \ge r/\cos(pi/n) $ or $\cos(\pi/n) \ge r/R $ or $\pi/n \le \cos^{-1} r/R $ or $n \ge \pi/\cos^{-1} r/R $.

For an inscribed $n$-gon in a circle of radius $R$, if $d$ is the distance to the center of the sides of the $n$-gon, $d/R =\cos(\pi/n) $. Since the inner circle can not be larger than this $d$, $r \le R\cos(\pi/n)$ or $\cos(\pi/n) \ge r/R $ or, again $\pi/n \le \cos^{-1}(r/R) $ or $n \ge \pi/\cos^{-1}(r/R) $.

So my bound is $n \ge \pi/\cos^{-1}(r/R) $.

marty cohen
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