Pick’s theorem

Let $P\subset\mathbb{R}^{2}$ be a polygon with all vertices on lattice points on the grid $\mathbb{Z}^{2}$. Let $I$ be the number of lattice points that lie inside $P$, and let $O$ be the number of lattice points that lie on the boundary of $P$. Then the area of $P$ is

 $A(P)=I+\frac{1}{2}O-1.$

In the above example, we have $I=5$ and $O=13$, so the area is $A=10\frac{1}{2}$; inspection shows this is true.

Title Pick’s theorem PicksTheorem 2013-03-22 12:46:58 2013-03-22 12:46:58 ariels (338) ariels (338) 4 ariels (338) Theorem msc 51A99 msc 05B99 msc 68U05