Graphene/hexagonal boron nitride (h-BN) heterostructures have attracted a great deal of attention in recent years due to their unique and complementary properties for use in a wide range of potential applications. However, it still remains a challenge to synthesize large-area high quality samples by a scalable growth method. In this work, we present the synthesis of both in-plane and stacked graphene/h-BN heterostructures on Cu foils by sequentially depositing h-BN via ion beam sputtering deposition (IBSD) and graphene with chemical vapor deposition (CVD). Due to a significant difference in the growth rate of graphene on h-BN and Cu, the in-plane graphene/h-BN heterostructures were rapidly formed on h-BN domain/Cu substrates. The large-area vertically stacked graphene/h-BN heterostructures were obtained by using the continuous h-BN film as a substrate. Furthermore, the well-designed sub-bilayered h-BN substrates provide direct evidence that the monolayered h-BN on Cu exhibits higher catalytic activity than the bilayered h-BN on Cu. The growth method applied here may have great potential in the scalable preparation of large-area high-quality graphene/h-BN heterostructures.