It is well-known that hyperbolic space is delta-hyperbolic, but what is delta?

Let $T$ be a triangle in $mathbb{H}^2$. Its area is $pi - alpha - beta - gamma$, where $alpha$, $beta$, and $gamma$ are the interior angles. You can find how slim this triangle is by considering an inscribed circle in $T$. The radius of this triangle, thus $delta$, are bounded above by the area, so to find the $delta$ that works for all triangles, you take the limit and consider an ideal triangle $T_infty$. You can explicitly compute that the inscribed circle minimizing distance between the sides has length $4 log phi$, where $phi$ is the golden ratio. (See here and here.)