Pick's theorem

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:

In the example shown, we have i = 7 interior points and b = 8 boundary points, so the area is A = 7 + 8/2 − 1 = 7 + 4 − 1 = 10 square units.

Note that the theorem as stated above is only valid for simple polygons, i.e., ones that consist of a single piece and do not contain holes. For a polygon that has h holes, with a boundary in the form of h + 1 simple closed curves, the slightly more complicated formula i + b/2 + h − 1 gives the area.

The result was first described by Georg Alexander Pick in 1899. The Reeve tetrahedron shows that there is no analogue of Pick's theorem in three dimensions that expresses the volume of a polytope by counting its interior and boundary points. However, there is a generalization in higher dimensions via Ehrhart polynomials. The formula also generalizes to surfaces of polyhedra.

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