Pick's Theorem

Georg Pick (1859 - abt 1942) was a mathematician who died in the Theresienstadt concentration camp. His simple geometry theorem allows us to find the area of any polygon that is embedded on a lattice or grid. It is especially useful for concave polygons, for which the usual textbook area formulas do not work.

Pick's Theorem works wonderfully on a geoboard, where students can design wild polygons with rubber bands stretched over the nailheads.

Pick's Theorem simply has us count up the number of points on the boundary of the polygon (where the rubber band touches a nailhead) and divide this number in half. Then add the number of points in the interior of the polygon (inside and not touching the rubber band) and subtract 1. We can write the formula like this: Area = B/2 + I - 1. For the polygon to the left, there are 16 boundary points and one interior point, so the area is 8/2 + 1 - 1 = 8 square units.

Geoboard dot paper|
Java applet
Picky nicky
Lesson plan