= Solution
The opposite face lies in the <plane> spanned by $\mathbf a,\mathbf b$ through the origin. Directly,
$$
\mathbf a\times\mathbf b=(-2,2,4),\qquad
\mathbf c\cdot(\mathbf a\times\mathbf b)=12,\qquad
|\mathbf a\times\mathbf b|=2\sqrt6.
$$
The <distance from a point to a plane> is the absolute normal component, so
$$
\boxed{d=\frac{12}{2\sqrt6}=\sqrt6}.
$$
Here $\mathbf c=(\mathbf a\times\mathbf b)/2$, so the perpendicular foot is the origin, a vertex of the opposite triangular face. Thus the distance to the actual face equals the distance to its containing <plane>.
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