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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?

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  • $\begingroup$ Integer/rational side as well I assume. $\endgroup$
    – fleablood
    Commented Feb 17, 2016 at 17:50
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    $\begingroup$ If you start with any regular pentagon $P$, and let $A$ be its area, then for any scalar multiple $rP$ the area is $r^2 A$. So just take $r=\sqrt{A}$ and you get area $1$. $\endgroup$
    – Lee Mosher
    Commented Feb 17, 2016 at 17:50
  • $\begingroup$ Presumably, you left out some condition (integer side length, integer vertices, integer something) $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Feb 17, 2016 at 17:52
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    $\begingroup$ @fleablood, ...if $a$ is an integer. Why should it be? $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Feb 17, 2016 at 17:55
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    $\begingroup$ Don't ####ing require people to follow links to understand your question!!!! If the question is important enough to post than it's important enough to include on the details in one place. Asking me to navigate to another post when I'm trying to help is both rude and lazy. DONT DO IT! Now edit this post and state the conditions!!!! $\endgroup$
    – fleablood
    Commented Feb 17, 2016 at 18:07

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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 $\pm1$.

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  • $\begingroup$ @gran_chat: So, definitely NOT is the answer; I was looking for a maybe "convenient" irrational radius giving a rational side (there are many examples). Consequently the "examples" in the web deserve well put its quotation marks. Thank you very much. $\endgroup$
    – Piquito
    Commented Feb 17, 2016 at 18:19

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