# Divide and Conquer: Proving Pick's Theorem for Lattice Polygons *

Difficulty | |

Time Required | Short (2-5 days) |

Prerequisites | None |

Material Availability | Readily available |

Cost | Very Low (under $20) |

Safety | No issues |

***Note:**This is an abbreviated Project Idea, without notes to start your background research, a specific list of materials, or a procedure for how to do the experiment. You can identify abbreviated Project Ideas by the asterisk at the end of the title. If you want a Project Idea with full instructions, please pick one without an asterisk.

## Abstract

If you like to play Tetris, then you might like this project. You will learn something interesting about the mathematics of complex shapes as you try to prove**Pick's Theorem**.

The strange shape below is an example of a *lattice polygon*, which is a polygon whose vertices lie on points in the plane that have integer coordinates.

As you can see, it is a complex shape, but there is an easy way to calculate its area, by simply counting lattice points!
If you count the number of lattice points on the boundary of the polygon (*b*), and the number of lattice points inside the polygon (*i*), then the area (*A*) of the polygon is given by Pick's Theorem:

**Equation 1: ** A = *i* + *b*/2 - 1.

**Figure 1.**Example of a lattice polygon.

A good way to explore lattice polygons is with a geoboard. A physical geoboard is a piece of wood with pegs (or nails) arranged in a regular grid. The wood represents a section of the plane, and the pegs or nails are the lattice points. You stretch rubber bands over the lattice points to create polygons. You can make or buy a geoboard for this project (it would make a nice addition to your display), but it is not absolutely necessary. You can also just draw a lattice and polygons on a piece of paper, or search online for a geoboard simulator.

Using whatever technique you choose, try experimenting with creating different polygon shapes, and see if Pick's theorem is correct. Start out with simple shapes and gradually move to more complex ones. Can you come up with a mathematical proof for Pick's Theorem?

## Objective

To study lattice polygons and prove that Pick's Theorem is correct.

## Share your story with Science Buddies!

Yes, I Did This Project! Please log in (or create a free account) to let us know how things went.## Credits

## Cite This Page

General citation information is provided here. Be sure to check the formatting, including capitalization, for the method you are using and update your citation, as needed.### MLA Style

*Science Buddies*, 28 July 2017, https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p009/pure-mathematics/picks-theorem-lattice-polygons. Accessed 24 June 2018.

### APA Style

*Divide and Conquer: Proving Pick's Theorem for Lattice Polygons.*Retrieved from https://www.sciencebuddies.org/science-fair-projects/project-ideas/Math_p009/pure-mathematics/picks-theorem-lattice-polygons

Last edit date: 2017-07-28

## Bibliography

This webpage has some information on Pick's Theorem, and a hint for how to go about proving it:

## News Feed on This Topic

*Note:*A computerized matching algorithm suggests the above articles. It's not as smart as you are, and it may occasionally give humorous, ridiculous, or even annoying results! Learn more about the News Feed

## Share your story with Science Buddies!

Yes, I Did This Project! Please log in (or create a free account) to let us know how things went.## Ask an Expert

The Ask an Expert Forum is intended to be a place where students can go to find answers to science questions that they have been unable to find using other resources. If you have specific questions about your science fair project or science fair, our team of volunteer scientists can help. Our Experts won't do the work for you, but they will make suggestions, offer guidance, and help you troubleshoot.Ask an Expert

## Related Links

## If you like this project, you might enjoy exploring these related careers:

### Mathematician

Mathematicians are part of an ancient tradition of searching for patterns, conjecturing, and figuring out truths based on rigorous deduction. Some mathematicians focus on purely theoretical problems, with no obvious or immediate applications, except to advance our understanding of mathematics, while others focus on applied mathematics, where they try to solve problems in economics, business, science, physics, or engineering. Read more### Math Teacher

Math teachers love mathematics and understand it well, but much more than that, they enjoy sharing their enthusiasm for the language of numbers with students. They use a variety of tools and techniques to help students grasp abstract concepts and show them that math describes the world around them. By helping students conquer fears and anxieties about math, teachers can open up many science and technology career possibilities for students. Teachers make a difference that lasts a lifetime! Read more## News Feed on This Topic

*Note:*A computerized matching algorithm suggests the above articles. It's not as smart as you are, and it may occasionally give humorous, ridiculous, or even annoying results! Learn more about the News Feed

## Looking for more science fun?

Try one of our science activities for quick, anytime science explorations. The perfect thing to liven up a rainy day, school vacation, or moment of boredom.

Find an Activity