Square - Expected R^2

Since $R$ is invariant under rotations about the origin, we may assume the sides of the square are parallel to the axes. Also, since the given square has $90^\circ$ symmetry about the origin, we may simply find the expected value of $R^2$ about one of the sides.

Now note that one of the sides is parametrized by $(x,y) = \left(X, \frac{1}{2}\right),$ that the assumption that the point is chosen uniformly over the perimeter means $X$ is uniformly distributed on $\left[-\frac{1}{2}, \frac{1}{2}\right],$ and that $R^2 = X^2 + \left(\frac{1}{2}\right)^2.$

Together, this means the expected value of $R^2$ is just $\mathbb{E}[R^2] = \mathbb{E}[X^2] + \frac{1}{4} = \int_{-1/2}^{1/2} x^2\,dx + \frac{1}{4} = \frac{1}{3} \approx \boxed{0.33333}$