Age 16 to 18 Challenge Level:
To discover the easy way to find this volume you can very quickly
make for yourself a model of the tetrahedron. Take a square $ABCD$
of stiff paper and fold it along the diagonal $AC$ . Open it until
the distance between $B$ and $D$ is equal to the length of the
sides of the square and then the four vertices are the vertices of
the tetrahedron with two equilateral faces and two isosceles faces.
You can either stand this 'tetrahedron' on an equilateral face (
$ABD$ or $CBD$ ) as its base or on an isosceles face ( $ABC$ or
$ADC$ ) as its base. One choice makes it easy to find the area of
the base and the height and so to find the volume.