Pick's Theorem

Let $A$ be the Area of a simply closed Polygon whose Vertices are lattice points. Let $B$ denote the number of Lattice Points on the Edges and $I$ the number of points in the interior of the Polygon. Then

$$A = I+{\textstyle{1\over 2}}B-1.$$

The Formula has been generalized to 3-D and higher dimensions using Ehrhart Polynomials.

See also Blichfeldt's Theorem, Ehrhart Polynomial, Lattice Point, Minkowski Convex Body Theorem

References

Diaz, R. and Robins, S. ``Pick's Formula via the Weierstraß $\wp$-Function.'' Amer. Math. Monthly 102, 431-437, 1995.

Ewald, G. Combinatorial Convexity and Algebraic Geometry. New York: Springer-Verlag, 1996.

Hammer, J. Unsolved Problems Concerning Lattice Points. London: Pitman, 1977.

Morelli, R. ``Pick's Theorem and the Todd Class of a Toric Variety.'' Adv. Math. 100, 183-231, 1993.

Pick, G. ``Geometrisches zur Zahlentheorie.'' Sitzenber. Lotos (Prague) 19, 311-319, 1899.

Steinhaus, H. Mathematical Snapshots, 3rd American ed. New York: Oxford University Press, pp. 97-98, 1983.