Volume of a prism

Question

Right rectangular prism has a base with diagonal a and a lateral face with diagonal b. Find the volume of the prism.

Answer 1

There is not enough information to answer the question. Consider two prisms with $2 \times 3$ bases, one with height $2$ and one with height $3$. The diagonal of the base is $\sqrt{13}$. The diagonal of the lateral face on the $3$ side of the first and the diagonal of the lateral face on the $2$ side of the second is also $\sqrt {13}$ but the volume of the first is $12$ and the second $18$.

Answer 2

Let the dimensions be $h,w, l$. The base diagonal $a=\sqrt{w^2 + l^2}$ and the lateral diagonal $b =\sqrt{h^2 + l^2}$. And the volume is $hwl$.

So $w = \sqrt{a^2 - l^2}$ and $h= \sqrt{b^2 - l^2}$ and Volume = $hwl = l\sqrt{(a-l)(a+l)(b-l)(b+l)}$

If $a,b$ are constants, then other than $0 < l < \min(a,b)$, $l$ is indeterminable. The volume may vary as $l$ varies.