The Hyperbolic Garden — an interactive mathematical observatory

The shape of a triangle

Curvature −1.00

Drag A, B, or C · or click to move the nearest pointNot drawn to Euclidean scale

Experiment I · Gauss–Bonnet

A triangle's angles reveal its area.

Move the three points. Their edges follow the shortest paths in hyperbolic space, and the angles always add up to less than 180°.

Curvature magnitude κ1.00

Flatter geometryStronger negative curvature

✦ A small expedition

Can you make the triangle's angle deficit exceed 2 radians? Push its corners toward the rim.

0.00 / 2.00 rad

Area = (π − α − β − γ) / κ²

Why does it work?

The disk is a model of the hyperbolic plane, with metric ds² = 4(dx² + dy²) / [κ²(1 − x² − y²)²]. A circle or line meeting the rim at right angles is a geodesic.

Its Gaussian curvature is −κ². Gauss–Bonnet says curvature integrated over a geodesic triangle plus its exterior-angle sum equals 2π. Rearranging gives κ²·Area = π − (α + β + γ).