Usually in hyperbolic geometry, a Gaussian curvature of $-1$ is assumed. However, if you imagine the surface of a pseudosphere with radius $\tfrac12$, then the gaussian curvature will become $-4$ per the formula $k = -1/r^2$ where $r$ is the radius of the pseudosphere.
Now, if I choose to map a triangle onto that surface, I assume that the triangle will have different angles and different area than the one on the pseudosphere of radius $1$.
Therefore, why do we only have the functions $\sinh,\,\cosh,\,\tanh$, that we've come to know, and not for other curvatures?