Spherical and hyperbolic geometries do not satisfy the parallel postulate.īy the way, 3-dimensional spaces can also have strange geometries. Another kind of non-Euclidean geometry is hyperbolic geometry. The geometry on a sphere is an example of a spherical or elliptic geometry. Planar geometry is sometimes called flat or Euclidean geometry. Here's a follow-up question for your students: are geodesic paths always the shortest paths between two points? Cool!Īnother neat fact about spherical triangles may be found in Spherical Pythagorean Theorem.ĭemonstrate the assertions about angles and areas with an example: draw a picture of a sphere and then draw a triangle whose vertices are at the north pole and at two distinct points on the equator.

Where the angles are measured in radians. On spheres, they correspond to pieces of great circles whose center coincide with the center of the sphere.)Ī triangle on a sphere has the interesting property that the sum of the angles is greater than 180 degrees! And in fact, two triangles with the same angles are not just similar (as in planar geometry), they are actually congruent! But wait, there's more: on a UNIT sphere, the AREA of the triangle actually satisfies:ĪREA of a triangle = (sum of angles) – Pi , (Such intrinsically straight lines are called ge ODEsics. However, triangles on other surfaces can behave differently!įor instance, consider a triangle on a sphere, whose edges are “intrinsically” straight in the sense that if you were a very tiny ant living on the sphere you would not think the edges were bending either to the left or right. Remember high school geometry? The sum of the angles of a planar triangle is always 180 degrees or pi radians.

Spherical geometry khan academy