Editing Hyperbolic geometry (section)

=== Triangles ===
{{main article|Hyperbolic triangle}}

Unlike Euclidean triangles, where the angles always add up to π [[radian]]s (180°, a [[straight angle]]), in hyperbolic space the sum of the angles of a triangle is always strictly less than π radians (180°). The difference is called the [[Angular defect|defect]]. Generally, the defect of a convex hyperbolic polygon with <math>n</math> sides is its angle sum subtracted from <math>(n - 2) \cdot 180^\circ</math>. 

The area of a hyperbolic triangle is given by its defect in radians multiplied by ''R''{{sup|2}}, which is also true for all convex hyperbolic polygons.<ref>{{Cite book |last=Thorgeirsson |first=Sverrir |url=https://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-227503 |title=Hyperbolic geometry: history, models, and axioms |date=2014}}</ref> Therefore all hyperbolic triangles have an area less than or equal to ''R''{{sup|2}}π. The area of a hyperbolic [[ideal triangle]] in which all three angles are 0° is equal to this maximum.

As in [[Euclidean geometry]], each hyperbolic triangle has an [[incircle]]. In hyperbolic space, if all three of its vertices lie on a [[horocycle]] or [[hypercycle (hyperbolic geometry)|hypercycle]], then the triangle has no [[circumscribed circle]].

As in [[spherical geometry|spherical]] and [[elliptical geometry]], in hyperbolic geometry if two triangles are similar, they must be congruent.