Given an $n$ dimensional closed bounded convex symmetric body $E$, situate it in $R^n$ so that the Euclidean ball $B$ is the ellipsoid of maximal volume contained in $E$. In 1978 Szarek, building on work of Kashin, showed (much more than) that if $({{vol(E)}\over{vol(B)}})^{1/n}\le C$, then the Banach space that has $E$ for its unit ball contains a subspace of dimension $n/2$ which is $C^2$ isomorphic to a Hilbert space. However, it is easy to see that $\ell_\infty^n$ contains a subspace well isomorphic to Hilbert spaces only of dimension of order $\log n$.
This is the most complicated answer I can think of to the original question but does show why someone might care about computing volume ratios.