The polyhedron volume 
can be computed for a closed polyhedron composed of
triangular faces with vertices 
using the divergence
theorem as
 |
where the normal 
is given by the cross product
 |
The vertices of each face must be ordered so that 
points outward. These formulas apply to polyhedra
with arbitrary faces, since faces
having more than three vertices can be triangulated.
They apply to convex polyhedra as well as nonconvex
polyhedra.
The volume can also be computed using the divergence theorem by integrating the vector field 
, which has divergence 
everywhere, over the triangulated faces of the
polyhedron.
