A method for manufacturing a sapphire substrate in which a plurality of projections are formed on a C-plane of the sapphire substrate by etching, includes: forming a patterned etching mask on the C-plane of the sapphire substrate; etching the sapphire substrate until the projections are formed, wherein each of the projections formed by the etching has a substantially triangular pyramidal-shape and has a plurality of side surfaces, a pointed top and a bottom, wherein the bottom of each of the projections has a substantially triangular shape having three outwardly curved arc-shaped sides. The projections are arranged on vertexes of a triangular lattice, and an orientation of the bottom of the projections conforms with an orientation that is rotated by about 30 degrees from an orientation of a triangle of the triangular lattice; and removing the etching mask from the sapphire substrate.