Tapering of vapour-liquid-solid (VLS) grown nanowires (NWs) is a widespread phenomenon resulting from dynamics of the liquid droplet during growth anddirect vapour-solid (VS) growth on the sidewall. To investigate both effects in ahighly controlled way, we developed a novel two-step growth approach for the selective area growth (SAG) of GaAs nanowires (NWs) by molecular beam epitaxy. In this growth approach optimum growth parameters are provided for thenucleation of NWs in a first step and for the shape variation during elongationin a second step, allowing NWs with a thin diameter (45 nm) and an untapered morphology to be realized with high vertical yield. We quantify the flux dependenceof radial VS growth and build a model that takes into account diffusion on theNW sidewalls to explain the observed VS growth rates. As our model is consistent with axial VLS growth we can combine it with an existing model for the diameter variation due to the droplet dynamics at the NW top. Thereby, we achieve fullunderstanding of the diameter of NWs over their entire length and the evolutionof the diameter and tapering during growth. We conclude that only the combinationof droplet dynamics and VS growth results in an untapered morphology. This result enables NW shape engineering and has important implications for doping of NWs.