Pick's theorem

Remarks

Both sides are additive when two lattice polygons are glued along a common edge, once the shared boundary points are counted correctly. So it suffices to check the formula for lattice triangles, and then for triangles with no lattice points other than their vertices, which have area $1/2$. Example: the triangle with vertices $(0, 0), (2, 0), (0, 2)$ has area $2$, $b = 6$ and $i = 0$, and $0 + 3 - 1 = 2$. Simplicity matters: for a region with holes the formula changes by a term for each hole. There is no direct analogue in dimension $3$: the tetrahedra with vertices $(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, r)$ have no other lattice points but volume $r/6$. The theorem is due to Georg Pick.

Going deeper

For a student

Pick's theorem links a continuous quantity, area, to a count of lattice points. It is the two-dimensional seed of Ehrhart theory, which counts lattice points in dilates of lattice polytopes. It also gives a quick proof that a lattice triangle with no other lattice points has area $1/2$, which is used in the study of Farey sequences.