Is it possible to express the length of a side of a regular $n$-gon (of unit circumradius) without transcendental functions? The question is exactly what the title says: 

Is it possible to express length of a side of a regular $n$-gon (of unit circumradius) without using transcendental functions (i.e.: sin, cos, tan,...)?

EDIT: It seems that the question is equivalent to solving any transcendental function such that x = 1/n, where n is a natural number, in terms of no transcendental functions.
 A: @Francesco is a dead ringer in his comment.  Since every cyclotomic polynomial equation is solvable by radicals, the same is true of the trigonometric functions that enter into the lengths of the sides.  So radical expressions exist for the side/cirumradius ratio in all regular polygons.
But there is some fine print.  If at any point in the process of solving the cyclotomic equation you need to solve a third or higher degree component, the radicals have complex arguments and you can't resolve the real quantities using algebra alone.  It's the famous casus irreducibilis in solving cubic equations, and more.  It happens with any irreducible prime degree equation if the degree is three or higher and multiple real roots can be expressed with radicals.  So in all cases you get radicals, but in most you can't solve them algebraically.
To get radicals that can be solved in terms of real numbers the cyclotomic equation must be expressible solely in terms of quadratic components.  That requires the degree to be a power of two.  Remember that the degree of the cyclotomic equation is the Euler totient function of the number of sides.  So we get radicals resolvable into real numbers if and only if the regular $n$-gon satisfies $\phi (n)=$ a power of two.  Thus $n \in \{3,4,5,6,8,10,...\}$.  Note that this is the same set as the set of constructible regular polygons.
