A triangle whose interior angles sum to zero
12 days ago
- #hyperbolic
- #spherical
- #geometry
- In spherical geometry, the sum of a triangle's interior angles exceeds π, and the area is determined by this excess.
- A sphere of radius 1 has triangle area equal to the angle sum minus π (Area = E = interior angle sum − π).
- Small spherical triangles have angle sums close to π; large ones can even have three right angles.
- In hyperbolic geometry, the sum of a triangle's interior angles is always less than π.
- For a space with curvature −1, the area equals the defect (Area = D = π − interior angle sum).
- Small hyperbolic triangles have angle sums near π, but as the sum approaches 0, the area approaches π.
- An improper hyperbolic triangle can have an angle sum of 0 and area π, with infinite perimeter but finite area.
- The radii of Euclidean half-circles in hyperbolic geometry do not affect the triangle's area.
- Spherical and hyperbolic geometries are locally Euclidean.