Special right triangle are in the ratio 1 : 2 : 3 and respectively measure 30° (), 60° (), and 90° (). The sides are in the ratio 1 : : 2. The proof of this fact is clear using trigonometry. The geometric proof is: The 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if "α", , are the angles in the progression then the sum of the angles = 180°. After dividing by 3, the angle must be 60°. The right angle
Hyperbolic triangle scale is most convenient if the lengths are measured in terms of the absolute length (a special unit of length analogous to a relations between distances in spherical geometry). This choice for this length scale makes formulas simpler. In terms of the Poincaré half-plane model absolute length corresponds to the infinitesimal metric formula_1 and in the Poincaré disk model to formula_2. In terms of the (constant and negative) Gaussian curvature of a hyperbolic plane, a unit of absolute length corresponds to a length of In a hyperbolic triangle the sum of the angles "A", "B", "C" (respectively opposite to the