The spontaneous assembly of chiral structures from building blocks that lack chirality is fundamentally important for colloidal chemistry and has implications for the formation of advanced optical materials. Here, we find that purified achiral gold tetrahedron-shaped nanoparticles assemble into two-dimensional superlattices that exhibit planar chirality under a balance of repulsive electrostatic and attractive van der Waals and depletion forces. A model accounting for these interactions shows that the growth of planar structures is kinetically preferred over similar three-dimensional products, explaining their selective formation.