This is a pretty straightforward application of Pick's Theorem. Since all the vertices of the $n$-gon are on the boundary, the area is at least $\frac{n}{2}-1$, which is equal to $\frac{n-2}{2}$. Since we still have to add the number of interior lattice points into our area, it is clear that the area will be at least $\frac{n-2}{2}$, since we cannot have a negative amount of lattice points.