Pick's Theorem

Revision as of 21:08, 16 October 2016 by Debussy (talk | contribs) (Proof)

Pick's Theorem expresses the area of a polygon, all of whose vertices are lattice points in a coordinate plane, in terms of the number of lattice points inside the polygon and the number of lattice points on the sides of the polygon. The formula is:

$A = I + \frac{B}{2} - 1$

where $I$ is the number of lattice points in the interior and $B$ being the number of lattice points on the boundary. It is similar to the Shoelace Theorem, and although it is less powerful, it is a good tool to have in solving problems.

[asy] size(150); defaultpen(linewidth(0.8)); for (int i = 1; i <= 5; i=i+1) { for (int j = 1; j <= 5; j=j+1) { dot((i,j)); } } draw((1,1)--(1,3)--(3,4)--(2,5)--(5,5)--(2,2)--(4,1)--cycle);[/asy]

Proof

{Outline: show that any triangle on the lattice points with no point in its interior or on its edges has an area of $\frac{1}{2}$. Then triangulate the shape and apply Euler's Polyhedron formula for graphs to obtain the desired result.}

Usage