Figure 1

A lattice point in the plane is any point that has
integer coordinates.
Let P be a polygon in the plane whose vertices have integer
coordinates. Then the area of P can be determined just
by counting the lattice points on the interior and boundary
of the polygon!
In fact, the area is given by
Area(P) = i + (b/2)  1
where i is the number interior lattice points, and
b is the number of boundary lattice points.
Presentation Suggestions:
A neat application of Pick's Theorem is that you cannot draw an equilateral triangle whose vertices lie on the lattice points.
Ask the students to try, making the point that the base does not need to be horizontal. Some convincing approximations may be obtained, but explain that none of them is really equilateral.
This is true because the area of an equilateral triangle of base A is an irrational multiple of A^{2}. If it is drawn on the lattice, then A^{2} is an integer use the Pythagorean theorem
and the area is irrational, contradicting Pick's theorem. (That's also a good opportunity to prove that the square root of 3 is irrational.)
The Math Behind the Fact:
Pick's theorem is nontrivial to prove. Start by
showing the theorem is true when there are no lattice
points on the interior.
How to Cite this Page:
Su, Francis E., et al. "Pick's Theorem."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
