linguistic:voice_change
Standardized Gaussian curvature

The relations among the angles and sides are analogous to those of spherical trigonometry; the length scale for both spherical geometry and hyperbolic geometry can for example be defined as the length of a side of an equilateral triangle with fixed angles.

The length 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 d s = | d z | Im (z) {\displaystyle ds={\frac {|dz|}{\operatorname {Im} (z)}}} and in the Poincaré disk model to d s = 2 | d z | 1 − | z | 2 {\displaystyle ds={\frac {2|dz|}{1-|z|^{2}}}}.

In terms of the (constant and negative) Gaussian curvature K 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 side with the corresponding letter) is strictly less than a straight angle. The difference between the measure of a straight angle and the sum of the measures of a triangle's angles is called the defect of the triangle. The area of a hyperbolic triangle is equal to its defect multiplied by the square of R:

This theorem, first proven by Johann Heinrich Lambert, is related to Girard's theorem in spherical geometry.