The volume-based scale distribution is the 3D extension of 2D surface-based scale distribution originally defined by Dumouchel et al. (2008). It allows characterizing the multi-scale features of shapes with complex morphology. It thus appears as an attractive metric for characterizing the primary atomization process where liquid structures are generally not spherical. On the other hand, curved surfaces such as liquid-gas interfaces can also be well represented by differential geometry and the use of intrinsic observables such as the two principal curvatures. In this study, we use results from differential geometry to build analytical bridges between the volume-based scale distribution and the geometry of the liquid-gas interface (namely the surface area, the mean and Gaussian curvatures). We also present some links between these quantities and the statistical moments of 'equivalent' systems constituted of either sheets, cylinder or droplets.