The shortest side must be $1$ or $2$. However none of $(1,1,6),(1,2,5),(1,3,4)$ form triangles, so the shortest side must be $2$. Then $(2,2,4)$ is degenerate, so the sides must be $(2,3,3)$.
This can be cut in half and reassembled into a rectangle with one side $1$ and diagonal $3$. By Pythagoras its area is then $2\sqrt2$ which means $\fbox{A}$