Pick's theorem (cf. http://en.wikipedia.org/wiki/Pick's_theorem): The area A of a simple polygon with all corners on a square grid is A = i + b/2 - 1 where i is the number of lattice points in the interior and b is the number of lattice points on the boundary. For the following A is the area as number of unit cells covered. Triangular lattice: A = 2*i + b - 2 Hexagonal lattice: A = i/2 + b/4 - 1/2 Square grid, every second column shifted by a half unit: A = i/2 + b/2 - 1 where b counts only the boundary points whose local neighborhood is not concave (i.e., 90 degrees of outside, 270 degrees of inside). I obtained these scribbling on a train ride yesterday. Best, jj P.S.: If anyone can access the following articles, then please email them to me. W. W. Funkenbusch,
From Euler's Formula to Pick's Formula using an Edge Theorem, The American Mathematical Monthly Volume 81 (1974) pages 647-648
Dale E. Varberg, Pick's Theorem Revisited, The American Mathematical Monthly Volume 92 (1985) pages 584-587 Branko Grünbaum and G. C. Shephard, Pick's Theorem, The American Mathematical Monthly Volume 100 (1993) pages 150-161