Spintronic devices using domain walls (DWs) as information carriers require stable high speeds and a clear understanding of their dynamics. The latter strongly depends on the geometry of the magnetic element hosting the DW. This work aims to determine the effects of the geometry on the width of a transverse DW nucleated in a bent magnetic nanowire with a rectangular cross-section. Our results show that a geometry-induced effective anisotropy is responsible for shrinking or stretching the DW width, depending on the direction along which the DW center points (DW phase).