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
Ailles rectangle between the two, with sides of lengths 2, 2 and (by the Pythagorean theorem) 2. The remaining space at the top of the rectangle is a right triangle with small angles of 15° and 75°. and sides of − 1, + 1 and 2. From the construction of the rectangle, it follows that and An alternative construction (also by Ailles) places a 30°–60°–90° triangle in the middle, flanked by two 45°–45°–90° triangles in the lower corners. The lower-left triangle has sides of 1, 1, and , while the lower-right triangle has sides of , , and . The 30°–60°–90° triangle