Given a 1x1 grid with 4 lattice points $[(0,0),(0,1),(1,0),(1,1)]$ (equivalent to a $2 \times 2$ grid of vertices), there are 2 shapes and areas that can be formed: a triangle and a square. There are 4 ways to obtain a $1/2u^2$ area (all the ways to form a triangle), while only one can result in a $1u^2$ area (since only one square can exist in the grid).
Assuming that no vertex of any of the shapes is found outside a lattice point, is there a way to calculate all possible areas and the amount of ways in which they can be formed for an $n \times n$ grid?