Pick's theorem
Sign in to saveformula that the area of a planar polygon whose vertices all have integer coordinates equals the number of interior integer points plus half the number of boundary integer points minus one
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Farey sunburst of order 6, with 1 interior (red) and 96 boundary (green) points giving an area of 1 + 96/2 − 1 = 48 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. It was popularized in English by Hugo Steinhaus in the 1950 edition of his book Mathematical Snapshots. It has multiple proofs, and can be generalized to formulas for certain kinds of non-simple polygons.
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