I am trying to inscribe polygons inside circles, given ONLY the lengths of the sides of the triangles. I have derived the angle, $\theta$, from the law of cosines, where $a$ and $b$ both equal the radius of the circle, and $c=d$ is the length of one side of the $n$-sided polygon: $$θ_d=arccos(1−\frac{d^2}{2r^2})$$ In the case of the "acute" polygon, a root finder for: $$\sum_{i=1}^narccos(1−\frac{{d_i}^2}{2r^2})=2\pi$$ is all that is needed to determine the radius, and then individual $\theta$'s.
However, if the polygon is "obtuse", the angle of the greatest length side will be $θ_d=2\pi-arccos(1−\frac{d^2}{2r^2})$, which represents the "other" solution to the inverse cosine (e.g. $arccos(.5)=\pm\frac{2\pi}3$).
It is important that I use the right equation prior to using a root-finder, as using the wrong equation causes undesired results in my calculations.
Is there a way to algebraicly determine whether the polygon fits the "obtuse" or "acute" categories using the lengths of the sides before running a program to calculate the radius?
Here is a picture of examples of my definition of acute and obtuse inscribed polygons.
