Hyperbolic Triangles

are triangles having -3- sides & -3- angles in the hyperbolic plane analogous to Euclidean    triangles.

the relations among the sides and / or angles are governed by those of spherical trigonometry.

Length of the sides is expressed in terms of a unit called   R  where R= 1/√(-K)1/2 , K    being Gaussian Curvature of the plane.

Each angle of the hyperbolic triangle, say ,A, B, C  is  slightly  less  than  corresponding  angle      say A',B',C'  in Euclidean triangle. Hence sum of the angles of a hyperbolic triangle is less than the sum of the straight angles in analogous Euclidean triangle & consequently less than π .

Mathematically, A+B+C<π and also A+B+C< A'+B'+C'. Difference  (A'+B'+C')-(A+B+C)=Defect

Area of hyperbolic triangle is given by πR2>area where area=defect*R2

In trigonometric relations involving the
sides-->hyperbolic functions are applied.
angles-->standard trigonometric functions are applied.
Further sinA/sinha =sinB/sinhb=sinC/sinhc

If C is the right angle in a hyperbolic right-angled triangle, and a is the side opposite to angle A, c is the side opposite to angle C and b is the side opposite to angle B, then

sinA= sinha/sinhc



The sides of a hyperbolic   triangle   can   become  as  large  as  possible but the area of the triangle is always less than π