---
tag: GCZ8EEDCHS
kind: concept
name: Hyperbolic triangle
msc: [51M10, 51M09]
sources:
  - url: https://en.wikipedia.org/wiki/Hyperbolic_triangle
same_as:
  wikidata: Q2246553
  wikipedia: Hyperbolic triangle
---

## Summary

A hyperbolic triangle is a triangle in the [[hyperbolic plane|GN4N94GQ0Z]]: three points not on one line, joined by geodesic segments.
Its angle sum is always less than $\pi$, and in curvature $-1$ its area equals the defect $\pi - (\alpha + \beta + \gamma)$.
So no hyperbolic triangle has area more than $\pi$.
Unlike in the Euclidean plane, a hyperbolic triangle is determined up to isometry by its three angles.
Its sides and angles are related by the [[hyperbolic law of cosines|7ZYKHQRKSJ]].

## Explanations

### L1 human

Draw a triangle on a saddle-shaped surface, with sides that are the shortest paths between the corners.
The sides bend inward compared with a flat triangle, so the corners become thinner and the three angles add up to less than $180^\circ$.
The bigger the triangle, the more angle it loses, and the lost angle measures its area.
In the limit, the three corners run off to infinity, the angles become $0$ and the area becomes exactly $\pi$; this is an ideal triangle.