This excellent solution came from Ruth from
Manchester High School for Girls.
In the tetrahedron $ABCD$, let $ABC$ and $ACD$ be right angled. If
you position the tetrahedron so that $ABC$ is the base, then the
vertex $D$ is directly above the edge $AC$. This means that the
height of the tetrahedron is the height of the triangle $ACD$ which
is $\frac{1}{\sqrt{2}}$ and the area of the base of the tetrahedron
is the area of the triangle $ABC$ which is $\frac{1}{2}$. The
volume of a pyramid is one third base times height. Therefore