Editing Hyperbolic triangle (section)

==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 [[hyperbolic geometry#Standardized Gaussian curvature|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.<ref>{{cite book|last=Needham|first=Tristan|title=Visual Complex Analysis|publisher=Oxford University Press|year=1998|isbn=9780198534464|page=270|url=https://books.google.com/books?id=ogz5FjmiqlQC&pg=PA270}}</ref>

In terms of the [[Poincaré half-plane model]] absolute length corresponds to the [[Riemannian manifold|infinitesimal metric]] <math>ds=\frac{|dz|}{\operatorname{Im}(z)}</math> and in the [[Poincaré disk model]] to <math>ds=\frac{2|dz|}{1-|z|^2}</math>.

In terms of the (constant and negative) [[Gaussian curvature]] {{mvar|K}} of a hyperbolic plane, a unit of absolute length corresponds to a length of
:<math>R=\frac{1}{\sqrt{-K}}</math>.

In a hyperbolic triangle the [[sum of angles of a triangle|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 [[angular defect|defect]] of the triangle. The [[area]] of a hyperbolic triangle is equal to its defect multiplied by the [[square (algebra)|square]] of&nbsp;{{mvar|R}}:
:<math>(\pi-A-B-C) R^2{}{}\!</math>.

This theorem, first proven by [[Johann Heinrich Lambert]],<ref>{{cite book|title=Foundations of Hyperbolic Manifolds|volume=149|series=Graduate Texts in Mathematics|first=John|last=Ratcliffe|publisher=Springer|year=2006|isbn=9780387331973|page=99|url=https://books.google.com/books?id=JV9m8o-ok6YC&pg=PA99|quotation=That the area of a hyperbolic triangle is proportional to its angle defect first appeared in Lambert's monograph ''Theorie der Parallellinien'', which was published posthumously in 1786.}}</ref> is related to [[Girard's theorem]] in spherical geometry.