As attractive analogue of graphene, boron monolayers have been theoretically predicted. However, due to electron deficiency of boron atom, synthesizing boron monolayer is very challenging in experiments. Using first-principles calculations, we explore stability and growth mechanism of various boron sheets on Cu(111) substrate. The monotonic decrease of formation energy of boron cluster B(N) with increasing cluster size and low diffusion barrier for a single B atom on Cu(111) surface ensure continuous growth of two-dimensional (2D) boron cluster. During growth process, hexagonal holes can easily arise at the edge of a 2D triangular boron cluster and then diffuse entad. Hence, large-scale boron monolayer with mixed hexagonal-triangular geometry can be obtained via either depositing boron atoms directly on Cu(111) surface or soft landing of small planar BN clusters. Our theoretical predictions would stimulate further experiments of synthesizing boron sheets on metal substrates and thus enrich the variety of 2D monolayer materials.