Pick's theorem

Description

Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon’s vertices are grid points, Pick’s theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon’s perimeter. ( The theorem is only valid for simple polygons, i.e., ones that consist of a single piece and do not contain “holes”)

Related formulas

Variables

AArea of the polygon (dimensionless)
IThe number of lattice points in the interior located in the polygon (dimensionless)
BThe number of lattice points on the boundary placed on the polygon's perimeter (dimensionless)