In mathematics, the term hyperbolic triangle has more than one meaning.
Hyperbolic geometry 
In hyperbolic geometry, a hyperbolic triangle is a figure in the hyperbolic plane, analogous to a triangle in Euclidean geometry, consisting of three sides and three angles. 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. In terms of the Gaussian curvature K of the plane this unit is given by
In all the trig formulas stated below the sides a, b, and c must be measured in this unit. 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 is often called the defect of the triangle. The area of a hyperbolic triangle is equal to its defect multiplied by the square of R:
Right triangles 
If C is a right angle then:
- The sine of angle A is the ratio of the hyperbolic sine of the side opposite the angle to the hyperbolic sine of the hypotenuse.
- The cosine of angle A is the ratio of the hyperbolic tangent of the adjacent leg to the hyperbolic tangent of the hypotenuse.
- The tangent of angle A is the ratio of the hyperbolic tangent of the opposite leg to the hyperbolic sine of the adjacent leg.
The hyperbolic sine, cosine, and tangent are hyperbolic functions which are analogous to the standard trigonometric functions.
Oblique triangles 
Whether C is a right angle or not, the following relationships hold.
There is a law of cosines:
a law of sines:
and a four-parts formula:
Ideal triangles 
If a pair of sides is asymptotic they may be said to form an angle of zero. In projective geometry, they meet at an ideal vertex on the circle at infinity. If all three are vertices are ideal, then the resulting figure is called an ideal triangle. An ideal hyperbolic triangle has an angle sum of 0°, a property it has in common with the triangular area in the Euclidean plane bounded by three tangent circles.
Euclidean geometry 
The length of the base of such a triangle is
and the altitude is
where u is the appropriate hyperbolic angle.
See also 
- Augustus De Morgan (1849) Trigonometry and Double Algebra, Chapter VI: "On the connection of common and hyperbolic trigonometry"
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