Hyperbolic Triangle Calculator

Calculations for a hyperbolic triangle. A hyperbolic triangle is a triangle on a hyperbolic plane. Like a spherical triangle, it is not part of Euclidean geometry (planar geometry). While the sum of the angles in a planar triangle is always 180 degrees, in a hyperbolic triangle, it is always less than 180 degrees.
Please specify the radius of curvature of the hyperbolic plane and the three side lengths of the hyperbolic triangle. The other values ​​will be calculated. The sum of the lengths of any two sides must always be greater than the length of the third side. The radius of curvature indicates the degree of curvature. The larger it is, the more the curvature approaches that of a plane. The calculation of the angles is derived from the hyperbolic law of cosines.

	Radius of curvature (r):
	Side a:
	Side b:
	Side c:
	Angle α:
	Angle β:
	Angle γ:
	Sum of angles α+β+γ:
	Perimeter (p):
	Area (A):
	Round to   0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15   decimal places.

Radius of curvature and sides have a one-dimensional unit (e.g. meter), the area has this unit squared (e.g. square meter).

A hyperbolic plane has constant negative curvature. Visually, it can be represented as a saddle-shaped surface. The concept of the hyperbolic plane as the foundation for a distinct geometry emerged from an examination of Euclid's parallel postulate, a statement that mathematicians had unsuccessfully attempted to prove or derive from the other Euclidean axioms for over 2000 years. The parallel postulate simplified states that, given a straight line and a point not on that line, there is exactly one line passing through that point that is parallel to the first line. This holds true on a flat plane. On a hyperbolic surface, however, there are infinitely many such parallel lines, whereas on an elliptic surface, there are none. Findings regarding hyperbolic geometry were first published around 1826 by the Russian mathematician Nikolai Lobachevsky, and independently around 1830 in a more general form by the Hungarian mathematician János Bolyai. Carl Friedrich Gauß also made similar discoveries around this time. Bernhard Riemann further developed these non-Euclidean geometries into the field of differential geometry, which Albert Einstein ultimately used to formulate his general theory of relativity.

Calculation of area and other dimensions: Hyperbolic Triangle

Last updated on 08/12/2026. Author: Jürgen Kummer

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