8 Pick’s Theorem

Pick’s theorem is a mathematical gem, showing that one can compute the area of a polygon

formed by vertices in the integer lattice simply by counting the number of vertex points in

the interior and on the boundary. How surprising! Let us explore this theorem and its

proof, which I find to be an excellent instance of how a difficult problem can be solved by

breaking it into easier instances. Allow me to present the theorem as a case study in the

advice of George P

´

olya:

If you cannot solve a problem, then there is an easier problem that you cannot

solve: find that problem. –P

´

olya (1973)

8.1 Figures in the integer lattice

To begin, consider the integer lattice, consisting of

the intersection points of the horizontal and vertical

integer lines in the plane, the points (a, b), where

both a and b are integers. By connecting such points

with line segments, we can form polygons in the

integer lattice; a simple polygon is one formed by

a closed-loop sequence of nonintersecting line seg-

ments in this way.

Theorem 63 (Pick’s theorem). The area of a simple polygon formed by vertices in the

integer lattice is precisely

A = i +

b

2

− 1,

where i is the number of lattice vertex points in the interior of the polygon and b is the

number of lattice vertex points on the boundary.

In the figure above, we have 11 interior points and 6 boundary points, and so the theorem

tells us that the area is 11 + 6/2 − 1 = 13.