Pick’s Theorem

Pick's theorem: Given a simple polygon whose vertices are all integers, Pick's theorem explains the relationship between its area A and the number of internal grid points i and the number of grid points on the edges b:

A = i + b/2 - 1

It can lead to some extension:

• Take the area of the figure composed of grid points as one unit. At the parallelogram lattice point, Pick's theorem still holds. Applied to any triangular lattice point, Pick's theorem is A = 2i + b - 2.

• For non-simple polygons P, Pick's theorem becomes A = i + b/2 - 𝛘(P), where 𝛘(P) represents the Euler eigennumber of P.

• High-dimensional generalization: Ehrhart polynomials

• Pick's theorem is equivalent to Euler's formula (V - E + F = 2).