Ehrhart Polynomials of Triangles in $\mathbb{R}^3$

We present two ways of constructing Ehrhart polynomials corresponding to certain triangles in $\mathbb{R}^3$. One of our methods employs Popoviciu's Theorem, while the other method uses a version of Pick's Theorem adapted to rank-2 lattices in $\mathbb{R}^3$. We then apply our methods to linear Diop