Recent advances in the self-assembly of block copolymers have enabled the precise fabrication of hierarchical nanostructures using low-cost solution-phase protocols. However, the preparation of well-defined and complex planar nanostructures in which the size is controlled in two dimensions (2D) has remained a challenge. Using a series of platelet-forming block copolymers, we have demonstrated through quantitative experiments that the living crystallization-driven self-assembly (CDSA) approach can be extended to growth in 2D. We used 2D CDSA to prepare uniform lenticular platelet micelles of controlled size and to construct precisely concentric lenticular micelles composed of spatially distinct functional regions, as well as complex structures analogous to nanoscale single- and double-headed arrows and spears. These methods represent a route to hierarchical nanostructures that can be tailored in 2D, with potential applications as diverse as liquid crystals, diagnostic technology and composite reinforcement.