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Concept# Pick's theorem

Summary

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.
Formula
Suppose that a polygon has integer coordinates for all of its vertices. Let i be the number of integer points interior to the polygon, and let b be the number of integer points on its boundary (including both vertices and points along the sides). Then the area A of this polygon is:
A = i + \frac{b}{2} - 1.
The example shown has i=7 interior points and b=8 boundary points, so its area is A=7+\tfrac{8}{2}-1=10 square un

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