What's the geometric meaning of the common value in the spherical law of sines?

Question

What's the geometric meaning of the common value in the spherical law of sines?

Hi,
On a unit sphere, a spherical triangle $ABC$ with angles $A, B, C$ and the opposite sides $a, b, c$.
The spherical law of sines reads:
$\displaystyle \frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c} = M$

My question is: What's the geometric meaning of the common value $M$?

Comments and thoughts:

1. Given the page Law_of_sines at Wikipedia, it implies:
$\displaystyle M^2 = (\frac{ V}{ sin(a)sin(b)sin(c)} )^2$ ,
where $V$ is the volume of the parallelepiped formed by the position vector of the vertices of the spherical triangle.
Then, the question is: what's the geometric meaning of $\displaystyle \sin(a)\sin(b)\sin(c)$ ?

2. The law of sines in hyperbolic geometry contains a link to an article Generalized Law of Sines which defines a generalized function $gsin(S)$, where $S$ can be a length $S_i$ in 1D and can be a triangle $S$ in 2D.

In 1D, $gsin(S_i)$ is defined, in the expression (1) in that article, as:
$\displaystyle gsin(S_i) = S_i - \frac{KS_i^3}{3!}+\frac{K^2S_i^5}{5!}-\frac{K^3S_i^7}{7!}+....$
And, $gsin(S_i)$ is $sin(S_i)$, $S_i$, $sinh(S_i)$ for Elliptic, Euclidean, Hyperbolic spaces, respectively.
So, let me say $gsin(S_i)$ has a geometric meaning in 1D.

In 2D, $gsin(S)$ is defned, in the expression (11) in that article, as:
$ gsin(S) = \lim_{K->K} \frac{polsin(S)}{K^{n/2}} = \lim_{K->K} \frac{polsinh(S)}{(-K)^{n/2}}$
I guess, $polsin(S)$ is polar sine psin() at Wikipedia which has a geometric meaning, this implies that $gsin(S)$ has a geometric meaning in 2D.

The expression (8) in that article implies:
$\displaystyle M = \frac{gsin(S)}{gsin(S_0) gsin(S_1) gsin(S_2)}$ .
Given that $gsin(S)$ and $gsin(S_i)$ has their geometric meanings, now the question is: what's the geometric meaning of $\displaystyle gsin(S_0) gsin(S_1) gsin(S_2)$ ? For the Elliptic case, this equals to: what's the geometric meaning of $\displaystyle \sin(S_0) \sin(S_1) \sin(S_2)$ ?