Block copolymers (BCPs) under nanoscale confinement can self-assemble to form novel nanostructures that are not available in the bulk state. Particularly, the ordering process of block copolymers and the resulting morphologies depend sensitively on the dimensionality, geometry, and surface property of the confining environment. In this study, we report on the self-assembled morphologies of polystyrene-block-1,4-polybutadiene (PS-b-PB) confined in conical pores of various sizes, shapes, and surface properties. Based on the experimental observations from transmission electron microscopy and theoretical calculation using the simulated annealing method, we found that the phase separation of PS-b-PB under the conical confinement is competitively determined by three thermodynamic factors: (1) the interfacial energy between two blocks, (2) the surface energy between the blocks and the surrounding environment (i.e., air and substrates), and (3) the entropic penalty associated with the large curvature at the vertices of conical pores. In addition, three-dimensional imaging of transmission electron microtomography was also performed in an attempt to gain more detailed information on the internal nanostructures of the BCP.