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 x base x 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: 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 $a^{2}b/6$.