Pick's theorem

A lattice polygon is a simple polygon whose corners are all points with integer coordinates. Suppose it has $I$ lattice points strictly inside and $B$ lattice points on its boundary.

Show that its area is
$$A=I+\frac{B}{2}-1.$$

You can cut the polygon into tiny triangles that contain no other lattice points, or reach for Euler's formula for polyhedra, which is also on the board. Is the second proof really independent of the first?