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How can i find the lenght of a side of a polygon with known number of sides that has a circle with known diameter inscribed in it? I'm a web-developer intereseted in this certain problem, that would be the solution to one of my aplications. And also is there a relation betwen a polygon that has a inscribed polygon with knwon distance betwen their sides? It would help even an answer for particular cases like pentagon or hexagon. I hope i've been specific enough :) enter image description here

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up vote 3 down vote accepted

For an $n$-gon the relations between circumscribed radius $R$, inscribed radius $\rho$ and side length $a$ are $$ \frac \rho R = \cos\frac\pi n $$ $$ \frac a R = 2\sin\pi n$$ $$ \frac a \rho = 2\tan\pi n.$$

For the second problem: If you have a polygon with side length $a$ and inscribed radius $\rho$, then the side length $a'$ for a smaller $n$-gon at distance $d$ is given by $$ \frac{a-a'}{a}=\frac d\rho$$ i.e. $$ a'=a\cdot\left(1-\frac d\rho\right).$$ Of course other data such as inscribed or circumscribed radius scale by the same factor $1-\frac d\rho$.

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You don't even need the "known distance". If the "known diameter" is $d$, then let the vertices of the pentagon be $A, B, C, D, E$ in counterclockwise order. Let $O$ be the center of the circle, and let $M$ be the midpoint of $AB$. Consider the right-angled triangle $AOM$; note that $\angle AOM = 360^\circ/10 = 36^\circ$. Hence, by trigonometry, $\overline{AM}/\overline{MO} = \overline{AM}/(d/2) = \tan(36^\circ)$. Use that to solve for $2\overline{AM}$, which is the side length of the outer pentagon.

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