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 🙂
[Math] How to 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
circlesgeometry
Best Answer
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 $1-\frac d\rho$.