Expected Distance of a Random Point from the Centre of a Regular Polygon

Pick a point uniformly at random in a regular $n$-gon with circumradius one. What is the expected distance to the centre? The Jacobian substitution $x = u, y = uv$ reduces the double integral to a tractable product, giving a closed form that evaluates to $0.460$ for $n=3$ and $0.541$ for $n=4$. The PDF includes three methods for sampling uniformly from a triangle (parallelogram, Kraemer, inverse CDF) and Python code verifying the closed form.