Associated Topics || Dr. Math Home || Search Dr. Math

Pick's Theorem

```
Date: 2/8/96 at 15:19:3
From: JimLindsay
Subject: Pick's theorem

Dear Dr. Math,

I was wondering what Pick's theorem is? Can you tell me?
Thanks so much!

Sincerely,
Brian Lindsay
```

```
Date: 2/8/96 at 16:22:33
From: Doctor Sarah
Subject: Re: Pick's theorem

Hi there -

There are some useful Web pages on Pick's theorem.  I'll include some
excerpts but you'll want to go look up the images yourself.
You can find more pages by using Alta Vista
http://altavista.digital.com/
to search the Web.

(1)
http://www.mcs.drexel.edu/~crorres/Archimedes/Stomachion/Pick.html

"Pick's theorem provides an elegant formula for the area of a simple
lattice polygon: a lattice polygon whose boundary consists of a sequence
of connected nonintersecting straight-line segments. The formula is

Area = I +B /2 - 1
where
I = number of interior lattice points
and
B = number of boundary lattice points

For example, the area of simple lattice polygon in the figure is

Area = 31 + 15 /2 - 1 = 37.5

The original result can be found in

Georg Pick
"Geometrisches zur Zahlenlehre"
Sitzungber. Lotos, Naturwissen Zeitschrift
Prague, Volume 19 (1899) pages 311-319.

Recent proofs and extensions of Pick's theorem can be found in

W. W. Funkenbusch
"From Euler's Formula to Pick's Formula using an Edge Theorem"
The American Mathematical Monthly
Volume 81 (1974) pages 647-648

Dale E. Varberg
"Pick's Theorem Revisited"
The American Mathematical Monthly
Volume 92 (1985) pages 584-587

Branko Grunbaum and G. C. Shephard
"Pick's Theorem"
The American Mathematical Monthly
Volume 100 (1993) pages 150-161"

(2)  http://134.121.112.29/fair_95/gym/hm022.html

"Pick's Theorem states that the area of any simple polygon whose
vertices are lattice points in the plane is given by this formula:

A=Vi + 1/2 Vb - 1

where Vi denotes the number of lattice points in the interior of the
polygon and Vb denotes the number of lattice points which occur in the
boundary between the vertices. See figure..."

There's more good information on methods in this paper.

Does this help?

-Doctor Sarah,  The Math Forum

```
Associated Topics:
College Definitions
College Triangles and Other Polygons

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search