We obviously would like to avoid computing an integral everytime we need a random variable. If we however switch to polar coordinates, we have for $x$ and $y$ $$\begin{equation*} r=\left(x^2+y^2\right)^{1/2} \hspace{1cm} \theta =tan^{-1}\frac{x}{y}, \end{equation*}$$ resulting in $$\begin{equation*} g(r,\theta)=r\exp{(-r^2/2)}drd\theta, \end{equation*}$$ where the angle $\theta$ could be given by a uniform distribution in the region $[0,2\pi]$. Following example 1 above, this implies simply multiplying random numbers $x\in [0,1]$ by $2\pi$.