1
$\begingroup$

Let $n$ be an integer. Find at least one $n$ such that the ratio between tha apothem and the side of a regular polygon with $n$ sides is an integer.

I found this problem while I was casually playing with some math.

I managed to get an expression for the wanted ratio, say $R$, that is $R = \frac{1}{2tan(\pi / n)}$, but here I am stucked.

$\endgroup$
1
$\begingroup$

Here (Corollary 5)

http://www.oberlin.edu/faculty/jcalcut/tanpap.pdf

you can find a proof that $\tan(\frac\pi n)$ is irrational for all $n>4$, so that the only rational value $R=\frac12$ is obtained for $n=4$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ thank you a lot!!! $\endgroup$ – Lorenzo Benedetti May 20 at 9:30
  • $\begingroup$ interesting reference (+1) $\endgroup$ – G Cab May 20 at 10:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.