TITLE: Angle of a regular simplex
QUESTION [22 upvotes]: I find the following question embarrassing, but I have not been able to either resolve it, or to find a reference.

What is the vertex angle of a regular $n$-simplex?

Background: For a vertex $v$ in a convex polyhedron $P$, the vertex angle at $v$ is the proportion of the volume that $P$ occupies in a small ball around $v$. In symbols, $$\angle v=\lim_{\varepsilon\to 0} \frac{|B(v,\varepsilon)\cap P|}{|B(v,\varepsilon)|}.$$ Up to normalization, this definition agrees with the familiar definition of the angles in the plane, or the solid angle in $3$-space.

REPLY [3 votes]: Here is a link to the original paper by Rogers mentioned by Joseph O'Rourke in his answer, with a short proof of the asymptotic formula for $F_n(\alpha)$, which is reproduced in the book Sphere Packings by Chuanming Zong.
It is also shown there that the multiplicative error in the formula is at most $1+ \frac{31}{12n} + O\left(\frac{1}{n^2}\right)$ in the case of the regular simplex, where $\sec 2\alpha=n$ (and thus $c=1$, or, $b=1$ in the paper).