Pick's Theorem

Date: 9/4/96 at 22:31:47
From: -={Mirage}=-
Subject: Pick's Theorem

Would you happen to know Pick's theorem? This has been bugging me for 
almost a year.

Thanks,
  Jeff

Date: 9/5/96 at 13:22:54
From: Doctor Tom
Subject: Re: Pick's Theorem

Hi Jeff,

Sure.

On a grid of points (like on graph paper) draw a simple polygon (no 
holes, no lines crossing each other), so that all the vertices lie on 
the grid. Then count the number of points completely inside (call this 
number I), and count the number of points that lie on the edges or 
corners of the polygon (call this number E). Then the area inside the 
polygon is given by the formula:  I + E/2 - 1, where the area is 
measured in terms of the smallest squares you can make on the grid.

You can find a proof of this in "Introduction to Geometry," second 
edition, by Coxeter (and probably in lots of other places - that's the 
first advanced geometry book I hauled off my shelf).

Try a bunch of examples yourself to convince yourself of the truth.  
The proof isn't too difficult either, but it's not totally elementary, 
so you may have to look it up.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/