First published by Georg Pick
in 1899, Pick's theorem
gives a surprising and elegant formula
for the area
of alattice polygon
(that is, a polygon drawn on a square grid so that each of its vertices
lies at the intersection of two grid lines).
Given such a shape, the apparently "easy" way to calculate its area is to split it into rectangles and triangles and sum their areas. Pick's beautiful result shows that even this is doing too much work: one need only count the number of lattice points (a "lattice point" is a point where two grid lines cross, that is, a corner of some square in the grid) lying within the shape and count the number lying on its boundary. If the number of interior points is I and boundary points B, then the area of the polygon is given by I+B/2-1 units squared (each square in the grid is assumed to have side length of one unit).
Interestingly, Pick's theorem is equivalent to Euler's famous formula regarding convex polyhedra.