Pick's theorem
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!