Hyperbolic triangle

A hyperbolic triangle is bounded by three hyperbolic geodesics and may have vertices on the ideal boundary.

Two corresponding side lengths and an equal nonincluded angle need not determine a hyperbolic triangle. Set $A=\pi /6$, $\cosh a=3/2$, $\cosh b=2$. The hyperbolic law of cosines has two distinct positive solutions $c=\ln (3\pm \sqrt{2})$. The two triangles share $a,b,A$ but have different third sides. An included angle instead determines the third side by the cosine rule.

Three corresponding side lengths determine a hyperbolic triangle up to an isometry of the hyperbolic plane. First match an ordered pair of vertices by an isometry. The third vertex lies at the intersection of two prescribed hyperbolic distance circles. There are at most two choices, interchanged by reflection in the geodesic through the matched pair; either choice can therefore be matched.

An ideal vertex lies on the boundary at infinity and has internal angle zero.

For curvature $-1$, a triangle of angles $\alpha ,\beta ,\gamma$ has area $\pi -\alpha -\beta -\gamma$ by Gauss--Bonnet.

For a geodesic hyperbolic $n$-gon of curvature $-1$ with interior angles ${\alpha }_1,\dots ,{\alpha }_n$, triangulation or the Gauss--Bonnet theorem gives

In curvature $-1$, a regular $n$-gon of circumradius $R>0$ has interior angle $\alpha (R)=2\arctan (\cot (\pi /n)/\cosh R)$, by the right-triangle angle form of the hyperbolic law of cosines. Thus its hyperbolic polygon area is $(n-2)\pi -n\alpha (R)$, continuously and strictly increasing from $0$ to $(n-2)\pi$. Every area in that open interval is therefore attained at exactly one circumradius.

For opposite side $a$, $\cosh a=(\cos \alpha +\cos \beta \cos \gamma )/(\sin \beta \sin \gamma )$.

A hyperbolic triangle is determined up to isometry by its three angles. Indeed, the angle form of the hyperbolic law of cosines determines every side length from the angles, after which hyperbolic side-side-side congruence determines the triangle. This differs from Euclidean geometry because hyperbolic space has an intrinsic length scale.