In the following, I use the spherical distance in $\mathbb{S}^n$.
Let $a,b,c,d\in\mathbb{S}^n$ and suppose we know the pairwise distances between them. Are there well known formulas for the maximal distance $M$ between any two points $x,y$ with $x\in[a,b]$ and $y\in[c,d]$? Or bounds for $M$ in terms of the distances between $a,b,c,d$? I am most interested in the case where $d(a,b),d(c,d)\leq\frac{\pi}{2}$ and the rest of the distances may be a bit bigger than $\pi/2$, but not much bigger.