What is the largest possible sum of all the angle measures of a $\Delta$ in hyperbolic space?

Question

$\Delta ABC$ exists in hyperbolic geometry. What is the maximum value for $m\angle A+m\angle B+m\angle C$?

Answer 1

$\pi- \epsilon$.

The sum of the angles of a hyperbolic triangle comes out to $\pi - Area(\Delta ABC)$, so by making the area close to zero, your angles will be close to $\pi$ (think of $\epsilon$ as a small constant).

On the other hand, by making a very large hyperbolic triangle, the angles approach zero and the sides are nearly parallel. (Try this in a model of hyperbolic space, like the Poincare disc.)

Answer 2

The area of a spherical triangle is given by $${\rm area}(\triangle)={\rm angle\ sum}-\pi\ ,$$ and for a hyperbolic triangle we have $${\rm area}(\triangle)=\pi-{\rm angle\ sum}(\triangle)\ .$$ Both formulas are a consequence of the famous Gauss-Bonnet theorem. It follows that in a hyperbolic triangle we have $${\rm angle\ sum}(\triangle)=\pi-{\rm area}(\triangle)\ .$$ As such triangles can have arbitrary small area $>0$ there is no hyperbolic triangle of maximal area, but we can say that the supremum of the angle sums is $\pi$. Note that small triangles of arbitrary shape can approximate this supremum.