Triangulation of $S^{1}\times S^{2}$

Question

I do not know if there are any answered question to it.
To construct the triangulation of $S^{3}$ is possible to use the fact that by taking two 3-balls and identifying their boundaries $S^{2}$, is possible to assign a tetrahedron to each sphere and then identify the two tetrahedra.
How can I think about $S^{1}\times S^{2}$?
$S^{2}$ can be decomposed into two disks, so that I can triangularize it with two triangles glued together?
What about $S^{1}\times S^{2}$: the triangulation is a tetrahedron with triangles ($S^{1}$) glued on each face or is something different?