A Relationship between Orthogonal Lattice Polyhedron and Pick’s Theorem

Abstract

Abstract. A lattice point is a point each of whose coordinates is integer. A lattice polygon is a polygon each of whose vertex is a lattice point. Pick’s theorem claims that the area M₂(P) of a lattice polygon P equals to I+B/2−1, where I is the number of lattice points contained in the inner region of P, and B is the number of lattice points contained in the boundary of P. In this paper, we give some formulas like Pick’s theorem, one of which calculates volume of a orthogonal polyhedron in R³ using ratio of solid angles.