According to my calculations, the average height of the unit hyper-hemisphere in $\mathbb{R}^n$ is given by the formula

$h_n:=\frac{S_{n-2}}{V_{n-2}} \int_{r=0}^1 r^{n-2} \sqrt{1-r^2} dr$,

where $S_{n-2}$, $V_{n-2}$ is respectively the surface and volume of the hypersphere in $\mathbb{R}^{n-1}$.

Question: How can we understand the quantity $h_n$? In particular, is $h_n$ increasing? Does it tend to $1$ as $n$ goes to infinity?