Elliptic geometry geometry is that the sum of the interior angles of a triangle is greater than 180 degrees. In the spherical model, for example, a triangle can be constructed with vertices at the locations where the three positive Cartesian coordinate axes intersect the sphere, and all three of its internal angles are 90 degrees, summing to 270 degrees. For sufficiently small triangles, the excess over 180 degrees can be made arbitrarily small. The Pythagorean theorem fails in elliptic geometry. In the 90°–90°–90° triangle described above, all three sides have the same length, and consequently do not satisfy formula_1. The Pythagorean result
Angular defect to less than 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to "less" than 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to "more" than 360°). In modern terms, the defect at a vertex or over a triangle (with a minus) is precisely the curvature at that point or the total (integrated) over the triangle, as established by the Gauss–Bonnet theorem. For a polyhedron, the defect at a vertex equals 2π minus the sum of all