I'm working on a problem for which I have to calculate the volume of a spherical triangle given the Solid Angle Ω. Specifically, if the Solid Angle Ω (in steradians) span by 3 vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ has been extracted using the well - known formula below
$\tan\frac{\Omega}{2} = \frac{|\vec{a}\cdot ( \vec{b} \times \vec{c} )|}{abc + (\vec{a}\cdot\vec{b})c + (\vec{c}\cdot\vec{a})b + (\vec{b}\cdot\vec{c})a}$,
what is the formula for the volume of a spherical triangle for Sphere radius $r$? I have seen many formulas including $V=\frac{\Omega}{3} r^3$ and $V = \frac{\Omega}{3}(r^3-(r-1)^3)$.
What is the correct formula and how does that come? If a have to add the volume of a spherical triangle and that of a pyramid for example, is there any need for normalization (for example as to 4π steradian)?
If $\Omega = 4 \pi$, i.e. the whole sphere, you want the volume to be the volume of the sphere $\frac43\pi r^3 = \frac\Omega 3 r^3$
The same is true for other solid angles representing a fraction of the sphere and so $\frac\Omega 3 r^3$ is correct
$\frac{\Omega}{3}(r^3-(r-1)^3)$ looks to me like the volume of a shell of depth $1$ and you need $r \ge 1$ for it to be meaningful