Ratios of Sizes of Mutually Inscribed Multidimensional Cubes and Spheres

Theorem
Consider:
 * a cube $C_n$ of $n$ dimensions inscribed within a sphere $S_n$ of $n$ dimensions


 * a sphere $S'_n$ of $n$ dimensions inscribed within a cube $'C_n$ of $n$ dimensions.

Let:
 * $A_{cn}$ be the $n$ dimensional volume of $C_n$
 * $A_{sn}$ be the $n$ dimensional volume of $S_n$
 * $A'_{cn}$ be the $n$ dimensional volume of $C'_n$
 * $A'_{sn}$ be the $n$ dimensional volume of $S'_n$.

For $n < 9$:
 * $\dfrac {S_n} {C_n} > \dfrac {C'_n} {S'_n}$

but for $n \ge 9$:
 * $\dfrac {S_n} {C_n} < \dfrac {C'_n} {S'_n}$

That is, for dimension $n$ less than $9$, the $n$ dimensional round peg fits better into an $n$ dimensional square hole than an $n$ dimensional square peg fits into an $n$ dimensional round hole, but for $9$ and higher dimensions, the situation is reversed.