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Pick's Theorem

by Eugene Chen of the Amador Valley Math Team

Pick's Theorem is another useful way to find the area of a polygon in the plane. It states that given a polygon in the coordinate plane, whose vertices are all lattice points, the area of the polygon is given by I+\frac{B}{2}-1, where I is the number of lattice points on the interior of the polygon, and B is the number of lattice points on the boundary of the polygon. This is illustrated by an example: 


Example: Find the area of the triangle below: 


 We count the boundary and interior lattice points: 


The blue dots indicate boundary lattice points and the red dots indicate interior lattice points. There are 
13
lattice points in its interior and 
6 
on its boundary, so the area is 
13+\frac{6}{2}-1=15
.



The proof will be left as a problem for the reader - hints are also given. 

Problem 1: Prove Pick's Theorem. 

Hints:
Use induction. 
The base case is to prove that Pick's Theorem works for triangles. 
For the inductive step, you want to show that you can split the polygon into two smaller polygons. 
You must prove that a polygon must have an interior diagonal. Construct one.