Sliced

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 $$(1/2)\times a\times b=(ab/2)$$ So using the formula for the volume of a pyramid: $$\text{Volume}=(1/3)\times (ab/2)\times (a/2)=a^{2}b/12$$ Times this by $2$ to get the volume for the big tetrahedron: $$(a^{2}b/12)\times 2=2a^{2}b/12=a^{2}b/6$$ So the volume of the big tetrahedron is $a^2b/6$.