# Contracting an angle (using straightedge and compass)

In my field theory lecture notes I have it that a regular polygon with $n$ sides is constructable iff $\zeta_{n}=\frac{2\pi}{n}$ is constructable.

Shouldn't this be $\frac{\pi}{n}$ instead of $\frac{2\pi}{n}$ ? (since each angle of a regular polygon with $n$ sides is $\frac{\pi}{n}$ and not $\frac{2\pi}{n}$)

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It doesn't matter, an angle $\theta$ is constructible if and only if $2 \theta$ is. – Will Jagy Sep 9 '12 at 22:30
What angle is in the regular $n$-gon is it you claim to be $\pi/n$? The only thing I can match it to is the angle between two neighboring diagonals at one of the corners. (Have you perhaps fooled yourself into thinking that the angle sum of every polygon is $\pi$? It is really $(n-2)\pi$). – Henning Makholm Sep 9 '12 at 22:52

Even more so in algebraic context, where the vertex set of a regular $n$-gon is just the set of $n$-th roots of unity...