- Christoph E.
Over a hundred years ago, Georg Alexander Pick (Austrian mathematician, 1859 - 1942) found an extraordinary formula for the area of simple lattice polygons ("simple": no intersecting line segments). Therefore, we have to determine the number of lattice points in the interior of the polygon (= I) and the number of lattice points on its boundary (= B). The area of the polygon (= A) equals I + B/2 - 1 (square units). Check this theorem by playing around with the applet which is dealing with a lattice pentagon. Move the big yellow points (= vertices of the pentagon) and compare the result given by Pick's formula to the area of the pentagon!
Hint: If you don't have confidence in the computed result (blue textbox), just build some ordinary polygons, e.g. squares, rectangles, triangles, ... What happens to the values if the polygon is not "simple"? Write down your conjecture!