The magnetic field dependence of the electronic properties of real single vertical quantum dots in circular and rectangular mesas is investigated within a full three-dimensional multiscale self-consistent approach without any á priori assumptions about the shape and strength of the confinement potential. Charging diagrams in a magnetic field for number of electrons up to five are also computed. Consistent with the experimental data, we found that the charging curves for the rectangular mesa dot in a magnetic field are flatter and exhibit less features than for a circular mesa dot. Evolution of the singlet-triplet energy separation in the two electron system for both dot geometries in magnetic field was also investigated. In the limit of large magnetic field, beyond the singlet-triplet transition, the singlet-triplet energy difference continues to become more negative in a circular mesa dot without any saturation whilst it is predicted to asymptotically approach zero for the rectangular mesa dot. This different behavior is attributed to the symmetry "breaking" that occurs in the singlet wave-functions in the rectangular dot.