In this study, the ratio between the end depth and the upstream depth which is known as the end depth ratio (EDR), is computed for the exponential (rectangular, triangular, and parabolic), the generalized trapezoidal (rectangular, triangular, semi-triangular, trapezoidal, semi-trapezoidal, inverted triangular and semi-inverted triangular) and the generalized circular (without horizontal base, with horizontal base at 4 different heights) channel cross-sections using different analytical methods for both sub- and super critical flow regimes, since these selected cross-sections are widely used in practice and also having experimental data sets for the comparison. Apart from that, based on the previously suggested theories for both sub- and super critical flow regimes the EDR and the end depth discharge (EDD) relationships for the above-mentioned cross-sections are as well obtained. As a novelty, two new approaches as well suggested as a part of this study that were only requiring the continuity and the energy equations; the three velocity point approach and the infinite number velocity points approach. These suggested approaches eliminate the need of the end pressure coefficient that was expected to be determined experimentally. These computed EDR and EDD values of the different theoretical approaches and the experimental data set of relevant cross-sections were statistically compared. Subsequently, using the brink depth, the direct discharge simple empirical relationships are generated for both flow regimes being a part of the main aim of this study, based on 4 different approaches that would be a toolkit for the engineers in practice in the relevant field. These proposed relationships are as well compared with their theoretically obtained results through the proper statistical measuring indices for their accuracies. Keywords: Brink, Circular, EDD, EDR, End depth, Exponential, Trapezoidal