ratio between volumes in $\mathbb{R}^n$

Question

Let $[-a_n,a_n]^n$ be the largest cube that fits into the n-sphere $S^{n-1}.$ Can we say what $a_n$ is? I mean, for $n=1$ we have $a_1=1$ and for $n=2$ we have $a_2 = \frac{1}{\sqrt{2}},$ so does this mean that $a_n = \frac{1}{\sqrt{n}}$? If yes, could anybody motivate how $a_n$ looks like for general $n$?

Answer 1

Hint: The point $(a_{n}, \dots, a_{n}) = a_{n}(1, \dots, 1)$ is a corner of the cube, and lies on the unit sphere. What is its distance from the origin in terms of $a_{n}$?