Is there a regular pentagon with integer area? Searching for an answer with a regular pentagon to the post I have unexpectedly the pertinent following question: **is there a regular pentagon with integer sides and integer area?** According to some references in the Web the answer could be affirmative (see this, that, and other where there are “examples” of (side, area)$=(16,440),(24,991),(6,60)$ respectively). However effective calculation gives, for example with side $24$, the area $990.9949….$ which is obviously not equal to $991$. I think the precedent examples are approximations and there is not a regular pentagon with integer area but I can not prove it so far. Some help or a counterexample?

If we require the pentagon to have integer side length, then the answer is no. The area of a regular pentagon with side length $a$ is $$5\cdot\left(\frac12a h\right)=\frac54a^2\tan(54^\circ)$$ But $\tan(54^\circ)$ is irrational, because the only rational values of $\tan k\pi/n$ are $0$ and $\pm1$.