Suppose that a regular $n$-gon has integer side length $m$. Is the lengths of its diagonals always algebraic numbers? If yes and if $n,m$ are given, is there an easy way to compute the diagonal lengths as a root of some polynomial?
Euclidean Geometry – Regular Polygon Diagonal Lengths
euclidean-geometry
Best Answer
All diagonals are either diameters, or sides of a triangle whose other two legs are segments uniting the center of the polygon to the diagonal's two extremities. As such, their lengths can be computed using the generalized Pythagorean theorem, also known as the law of cosines. The angle between these two sides is a rational multiple of π, hence its sine and cosine are both algebraic, being the real and imaginary part of the n complex roots of unity, per de Moivre's and Euler's formula. Now all that's left to show is that these radii are themselves algebraic: Indeed, $r^2+r^2$$-2rr\cos\dfrac{2\pi}n=m^2$ implying $r=\dfrac m{\sqrt{2\Big(1-\cos\frac{2\pi}n\Big)}}\in\mathbb{A}$. Then $d_j=r\sqrt{2\bigg(1-\cos\dfrac{2k\pi}n\bigg)}\in\mathbb{A}$. QED.