Extension of Pick’s Theorem to Spherical Geometry using Girard’s Theorem

Abstract

In this article, Pick’s theorem is extended to three-dimensional bodies with two-dimensional surfaces, namely spherical geometry. The equation for the area of a polygon consisting of equilateral spherical triangles is obtained by combining Girard’s theorem used to find area of any spherical triangle and Pick’s theorem used to find area of a simple polygon with lattice point vertices in Euclidian geometry. Vertices of the polygon are represented by integer points. In this way, an equation to find area of a spherical polygon is presented. This equation could give an idea to be applied on cylindrical surfaces, hyperbolic geometry and more general surfaces. The theorem proposed in this article which is the extension of Pick’s theorem using Girard’s theorem seems to be a special case of a more general theorem.