TITLE: Volume of tetrahedron using cross and dot product
QUESTION [14 upvotes]: Consider the tetrahedron in the image:  Prove that the volume of the tetrahedron is given by $\frac16 |a \times b \cdot c|$.

I know volume of the tetrahedron is equal to the base area times height, and here, the height is $h$, and I’m considering the base area to be the area of the triangle $BCD$. 
So, what I have is:
$$\begin{align}
\text{base area} &= \frac12 \lvert a \times b \rvert \\
\text{height $h$} &= \lvert c\rvert \cos \theta
\end{align}$$
So volume is $$V=\frac12 \lvert a \times b\rvert \cdot \lvert c\rvert \cos \theta $$
But I don’t know how to arrive from this at $\frac16 |a \times b \cdot c|$.
Please advise.

REPLY [3 votes]: Hint: $\mathbf{a}\times\mathbf{b}$ "points straight up".  It is therefore parallel to the line $h$ that you show.  Therefore, it has the same angle with $c$ as $h$ does.