Pick's Formula

Date: 05/22/97 at 07:34:45
From: Amy Lee
Subject: Area of an irregular shape or object.

What is Pick's Law? Can you please explain it to me? 

Date: 05/23/97 at 10:09:00
From: Doctor Jerry
Subject: Re: Area of an irregular shape or object.

Hi Amy, 

Pick's Law is difficult to explain in detail, but I'll give an 
example.

Draw a coordinate system and label the "lattice points" (0,0), (1,0),
(2,0), (3,0), (0,1), (1,1), (2,1), (3,1), (0,2), (1,2), (2,2), (3,2), 
(0,3), (1,3), (2,3), and (3,3).  

Draw a polygon by connecting the points (3,0), (3,1), (3,2), (3,3), 
(2,3), (2,2), (1,2), (1,1,), and (3,0). 

By counting squares, the area enclosed by this polygon is 4.5 units.

Pick's Formula gives another way of calculating this area.  Pick's 
Formula is:

  A = I +(1/2)*B - 1.

I is the number of lattice points inside the polygon, not touching any 
of the sides. In this case there's just one lattice point inside, 
namely, (2,1). So, I = 1. B is the number of lattice points on the 
edges of the polygon. If you count, you will find that B = 9.  So,

  A = 1 + (1/2)*9 - 1 = 1 +4.5 - 1 = 4.5

Pick's formula is not easy to prove.

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