Pick's theorem

Pick's theorem gives a simple way to find the area of a polygon whose vertices are all points with integer coordinates (lattice points). The formula relates the area to the number of lattice points inside the polygon and on its boundary.

Important properties

• The polygon must be simple (no holes, no self-intersections) and have all vertices at lattice points.

• Let $I$ be the number of lattice points strictly inside the polygon.

• Let $B$ be the number of lattice points exactly on the boundary.

• The area $A$ of the polygon is given by Pick's theorem: $A = I + \frac{B}{2} - 1$

• Works only for polygons with vertices at integer coordinates.