Pick’s theorem

Let $P\subset {ℝ}^2$ 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
Canonical name	PicksTheorem
Date of creation	2013-03-22 12:46:58
Last modified on	2013-03-22 12:46:58
Owner	ariels (338)
Last modified by	ariels (338)
Numerical id	4
Author	ariels (338)
Entry type	Theorem
Classification	msc 51A99
Classification	msc 05B99
Classification	msc 68U05