When is area of regular $n$-gon expressible by radicals?

Question

I had a look at the table here. As there are only $20$ values published I noted that the area values given as radicals ( of square roots ) correspond to values of $n$ for which the $n$-gon is constructible. But I am now surprised that for the values $n=15$ and $n=17$ whose $n$-gons are constructible there is no expression with radicals given.

Why is that the case ? Is a general theorem known about the phenomenon ? What are the values of $n$ for which the area of regular $n$-gon is expressible by radicals ?

Answer 1

But I am now surprised that for the values $n=15$ and $n=17$ whose $n$-gons are constructible there is no expression with radicals given. Why is that the case ?

Probably because of the fact that they would look something like this:

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When I understand correctly, the value of $n=17$ is simply missing because the expression is far too long and complicated... ;-$)$

Yeah, one might be slightly tempted to say that...