Let \(\int_a^b f(x)w(x)dx \approx w_0f(a)+\sum_{j=1}^N w_j f(x_j)+\delta w_{N+1}f(b)\) be the classical Gauss-Radau (\(\delta=0\)) or the Gauss-Lobatto (\(\delta=1\)) quadrature formulae. This note is devoted to study the connection between the boundary weight \(w_0\) associated with the fixed node \(x_0=a\) and weights \(w_j\) corresponding to interior nodes \(x_j\). Some general expressions for the weights of the Gauss-Radau quadrature are obtained in terms of the eigenvalues of the Sturm-Liouville problem associated to orthogonality. As an application of the former results, it is obtained a characterization of the Gauss-Radau quadrature in terms of a one-point rule.