Question about Pick's Theorem

Solution 1:

Certainly there is. One such version is stated as Theorem 4.1 in this paper of mine:

Theorem (Pick's Theorem): Let $\Lambda$ be a two-dimensional lattice in $\mathbb{R}^k$ with 2-volume $\delta$. Let $P$ be a $\Lambda$-lattice polygon containing $h$ interior lattice points and $b$ boundary lattice points. Then the area $A(P)$ of $P$ is equal to $\delta \cdot (h + \frac{b}{2} - 1)$.

As a reference to the proof I give a 2003 book of Erdős and Surányi. But -- especially for the $k = 2$ case that you asked about -- Jim Conant's comment is right on: the proof consists simply of making a linear change of variables to get from $\Lambda$ to $\mathbb{Z}^2$.