This paper deals with the computation of integrals of centred bivariate Gaussian densities over any domain defined as an angular sector of $\mathbb{R}^2$. Based on an accessible geometrical approach of the problem, we suggest to transform the double integral into a single one, leading to a tractable closed-form expression only involving trigonometric functions. This solution can also be seen as the angular cumulative distribution of bivariate centered Gaussian variables $(X,Y)$. We aim to provide a didactic approach of our results, and we validate them by comparing with those of the literature.