Pick’s formula is a way to find the area of a geoboard polygon by counting interior pegs and boundary pegs. Students can discover the formula by doing some experimenting under teacher guidance (see Geometry Labs 8.6 or Algebra: Themes, Tools, Concepts 4.12.) I have used this in the classroom for decades, because it is such a beautiful and surprising result, which provides a wonderful connection between algebra and geometry, and because it is both accessible and challenging in just the right way. But it’s only three years ago, at a meeting of Escape from the Textbook! that I started to understand the underlying math.
Read all about this on a new page, for teachers, on my Web site: Proving Pick’s Formula. There, I provide an illustrated step-by-step proof of why the formula will always work. I leave some of the algebra for you to do, so get a pencil and paper before clicking.
Many thanks to Kim Seashore, Avery Pickford, and Dan Bennett for their help with this!