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  "body": "\n## Summary\n\nA hyperbolic triangle is a triangle in the [[hyperbolic plane|GN4N94GQ0Z]]: three points not on one line, joined by geodesic segments.\nIts angle sum is always less than $\\pi$, and in curvature $-1$ its area equals the defect $\\pi - (\\alpha + \\beta + \\gamma)$.\nSo no hyperbolic triangle has area more than $\\pi$.\nUnlike in the Euclidean plane, a hyperbolic triangle is determined up to isometry by its three angles.\nIts sides and angles are related by the [[hyperbolic law of cosines|7ZYKHQRKSJ]].\n\n## Explanations\n\n### L1 human\n\nDraw a triangle on a saddle-shaped surface, with sides that are the shortest paths between the corners.\nThe sides bend inward compared with a flat triangle, so the corners become thinner and the three angles add up to less than $180^\\circ$.\nThe bigger the triangle, the more angle it loses, and the lost angle measures its area.\nIn the limit, the three corners run off to infinity, the angles become $0$ and the area becomes exactly $\\pi$; this is an ideal triangle.\n",
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