In researching a kind of geometric problem I have found the following integer parameter dependent constant, geometrically describing the circumradius of some regular $m$-gon:
$$ \rho(m,\mu) = \frac{\cos\left(\frac{\pi}{m}+\mu\frac{\pi}{m}\right)}{2\sin\left(\frac{\pi}{m}\right)\sin\left(\mu\frac{\pi}{m}\right)} $$
Here, $m>4$ and $\left\lceil m/2 \right\rceil-1 > \mu > 0$. The latter condition guarantees that $\rho(m,\mu)$ is positive.
Now, I have observed that $\rho(m,\mu)$ sometimes can be obtained as the absolute value of a sum of complex roots of unity $\omega_{m}^{k} = \exp(i 2 \pi k / m)$ (= basic vectors of unit length in the $m$ orientations of the $m$-gon). Namely,
$$ \rho(m,\mu) = \left|\sum_{k=0}^{m-1}c_{k}\omega_{m}^{k}\right| $$
with non-negative integer coefficients $c_{k}$. For instance,
$$\rho(8,1) = \left|\omega_{8}^{1}+\omega_{8}^{0}+\omega_{8}^{7}\right|$$
$$\rho(10,1) = \left|\omega_{10}^{0}+2\omega_{10}^{1}+2\omega_{10}^{9}\right|$$
$$\rho(10,2) = \left|\omega_{10}^{1}+\omega_{10}^{9}\right|$$
$$\rho(10,3) = \left|\omega_{10}^{2}+\omega_{10}^{8}\right|$$
$$\rho(12,1) = \left|2\omega_{12}^{1}+3\omega_{12}^{0}+2\omega_{12}^{11}\right|$$
$$\rho(12,2) = \left|\omega_{12}^{1}+\omega_{12}^{0}+\omega_{12}^{11}\right|$$
Thus, the problem is, geometrically, when can the circumradius vector of a regular $m$-gon of a given length be constructed from the m-th roots of unity interpreted as $m$ basic vectors in the complex plane?
However, trying to find examples for all admissible parameters according to the aforementioned conditions by some exhaustive computer search was futile. In particular, I did not find any solutions for odd $m$, but also not always a full set of solutions for even $m$ and all possible values of $\mu$. I guess, there are no solutions for all possible parameter choices of $m$ and $\mu$.
My question is, can one show/proof that only certain combinations of parameters have such a construction by complex roots of unity? And maybe find all the possible constructions? And maybe explain, why the other cases are impossible?
I have no idea, how to proceed. One observation I made is that sums of complex roots of unity appear as roots of abelian polynomials, but I don't know if that helps?
One observation is that the absolute value of a sum of roots of unity is an algebraic integer. If $m$ and $\mu$ are integers, $\rho(m,\mu)$ is an algebraic number, but if is not an algebraic integer (i.e. if it is a root of a monic irreducible polynomial whose coefficients are rational but not all integers), then it is not the absolute value of a sum of roots of unity. For example, $\rho(7,1)$ is a root of $z^3 - z^2 - z -1/7$, and therefore is not an algebraic integer.