I'm looking for a way to calculate the volume of a cut-off simplex. That is, say we have a (n-1) convex simplex in the n-dimensional case, and that the simplex is cut in two parts. For example, in the three-dimensional case, the simplex is a triangle and then we cut it into two segments. There are different ways of cutting the triangle, we can either get two triangles, or in the other case a triangle and a convex polytope with 4 vertices. I'm looking for a way of calculating the individual content (volume) of these two segments.

Can anyone point me to a reference of how to do this?

Thanks!