How small can the sum of the interior angles of a triangle in hyperbolic geometry get? Is there a lower bound?

Question

I know that in Euclidean geometry, the sum of the interior angles of a triangle is exactly $\pi$.

In hyperbolic geometry, I know that the sum of the interior angles of a triangle is $\leq \pi$, and I know that there exist triangles in hyperbolic geometry with interior angles that sum to strictly less than $\pi$.

However, what I don't know is ...

How small can the sum of the interior angles of a triangle in hyperbolic geometry get? Is there a lower bound?

Is there a hyperbolic triangle with interior angle sum equal to, say, $1/10000000$? I suspect it is something nice like the angles have to add up to be at least $\pi/2$ but I don't know if that's true.

Answer 1

(Sorry if this should be a comment and not an answer: please advise). There is no lower bound. See, for example, Existence of triangles with three arbitrarily small angles in Archimedian Neutral Geometry, (e.g. in hyperbolic geometry). From baby Hartshorne for a rough sketch of a synthetic proof.

Answer 2

Comment

Equilateral triangle on Poincare disc, each vertex zero (spike) angle.

Angle defect$\pi.$