Daniel, from Wales High School, sent us
this very elegant solution:
The formula for the volume of a tetrahedron is $$1/3 \times
\text{area of base} \times \text{perpendicular height}$$ If you
slice the tetrahedron in half through $b$ you end up with two equal
smaller pyramids. I will work out the volume of one and then
multiply by $2$ because they have equal volumes. So to start, I
must work out the area of the base. The area of a triangle is
calculated as follows: $$1/2 \times \text{base} \times
\text{perpendicular height}$$ The base is $a$ and the perpendicular
height is $b$ so the area of the base is $$(1/2)\times a\times b=(a
b/2)$$ So using the formula for the volume of a pyramid:
$$\text{Volume}=(1/3) \times (ab/2)\times (a/2)=a^2b/12$$ Times
this by $2$ to get the volume for the big tetrahedron:
$$(a^2b/12)\times 2=2a^2b/12 =a^2b/6$$ So the volume of the big
tetrahedron is $a^2b/6$.