Pick's Theorem gives the area of a simple polygon whose vertices all lie on points of an integer lattice, expressing that area in terms of the number of lattice points strictly inside the polygon and the number of lattice points on its boundary. Proved by Georg Alexander Pick, it provides an exact area formula requiring only counting, with no measurement of lengths or angles.
Facts
StatementFor a simple polygon whose vertices all lie at integer lattice points, the enclosed area A equals i plus half of b minus 1 (A = i + b/2 - 1), where i is the number of lattice points strictly interior to the polygon and b is the number of lattice points on its boundary. 1 Proof YearYear the result was first described by Georg Alexander Pick; popularized in English by Hugo Steinhaus in the 1950 edition of Mathematical Snapshots. Classification
Statement Form Connections
Sources
1. Pick's Theorem (Wikipedia)
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In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1899.
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