:not to be confused with the hyperbolic triangle of a hyperbolic sector in hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. it consists of three line segments called sides or edges and three points called angles or vertices. just as in the euclidean case, three points of a hyperbolic space of an arbitrary dimension always lie on the same plane. hence planar hyperbolic triangles also describe triangles possible in any higher dimension of hyperbolic spaces.  ==definition== a hyperbolic triangle consists of three non-collinear points and three segments between them., interactive instructional websitesvetlana katok (1992) fuchsian groups, university of chicago press isbn 0-226-42583-5 what namely says the book on the definition? --> the relations among the angles and sides are analogous to those of spherical trigonometry; they are most conveniently stated if the lengths are measured in terms of a special unit of length analogous to a radian.for instance, the absolute length scales for both spherical geometry (the radian) and hyperbolic geometry can be defined as the perimeters of equilateral triangles with fixed angular defects; see . as with the radian, the choice of units for this length scale is the one that makes the area formula as simple as possible. in terms of the gaussian curvature  of the plane this unit is given by ::r=\frac{1}{\sqrt{-k}}. 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. this is contrasted to euclidean triangles where this sum is always equal to the straight angle, as well as to spherical triangles where this sum is greater. the difference is often called the defect of the triangle. the area of a hyperbolic triangle is equal to its defect multiplied by the square of&nbsp;: ::(\pi-a-b-c) r^2{}{}.\! this theorem, first proven by johann heinrich lambert,{{citation|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=http://
Hyperbolic_geometry Triangles