Explanation and Test Case for Pick's Theorem

Date: 06/13/2006 at 02:47:54
From: Anelisa
Subject: the use of area theorems to check validity of Pick's theorem

I want to be able to test Pick's Theorem for the area of polygons.  I
want to accompany each polygon with a detailed test of the formula.

But I don't really understand Pick's Theorem and its formula.  Can you
explain the formula and show that it works for a polygon?

Date: 06/13/2006 at 12:51:01
From: Doctor Peterson
Subject: Re: the use of area theorems to check validity of Pick's theorem

Hi, Anelisa.

If what you want to do is just to check the formula in specific 
cases, I would draw a few lattice polygons and use both methods to 
find the area. For example, consider the quadrilateral ABCD:

  A---o---B   o
   \        \
  o \ o   o   C
     \    /
  o   D   o   o

By Pick's rule, A = I + B/2 - 1, where I is the number of Interior 
lattice points, and B is the number of Boundary lattice points.  We 
have I=2 since there are two dots entirely inside the polygon, and 
B=5, since besides the four vertices there is one extra dot on the 
boundary.  So

  A = 2 + 5/2 - 1 = 3 1/2

How can we find the area using ordinary area formulas?  We'll have to 
break it up into simple triangles (or, sometimes, rectangles or 
parallelograms) whose base and height we know.  Having just mentioned 
parallelograms, I see one neat way to break this up:

  A---o---B   o
   \\       \
  o \ X---o---C
     \|   /
  o   D   o   o

We have one parallelogram and two triangles:

  ABCX has base 2 and height 1, so A = 2*1 = 2
  AXD has base 1 (namely XD) and height 1, so A = 1*1/2 = 1/2
  CXD has base 2 (namely XC) and height 1, so A = 2*1/2 = 1

Adding these up, the total area is 2 + 1/2 + 1 = 3 1/2, which agrees 
with Pick.

Demonstrating that it works in a few examples doesn't prove the 
formula, of course; in our archives we have several discussions of 
how to prove it.

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/