Remember high school geometry? The sum of the angles of a planar triangle is always 180 degrees or Pi radians. However, triangles on other surfaces can behave differently!

For 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. (Such intrinsically straight lines are called *geodesics*. On spheres, they correspond to pieces of great circles whose center coincide with the center of the sphere.)

A 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:

AREA of a triangle = (sum of angles) – Pi ,

where the angles are measured in radians. Cool!

Another neat fact about spherical triangles may be found in Spherical Pythagorean Theorem.

**Presentation Suggestions:**

Demonstrate 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. Here’s a follow-up question for your students: are geodesic paths always the shortest paths between two points?

**The Math Behind the Fact:**

Planar geometry is sometimes called *flat* or *Euclidean* geometry. The geometry on a sphere is an example of a *spherical* or *elliptic* geometry. Another kind of non-Euclidean geometry is hyperbolic geometry. Spherical and hyperbolic geometries do not satisfy the parallel postulate.

By the way, 3-dimensional spaces can also have strange geometries. Our universe, for instance, seems to have a Euclidean geometry on a local scale, but does not on a global scale. In much the same way that a sphere is “curved”, so that divergent geodesics extending from the south pole will meet again at the north pole, Einstein suggested that 3-space is “curved” by the presence of matter, so that light rays (which follow geodesics) bend near very massive objects!

**How to Cite this Page:**

Su, Francis E., et al. “Spherical Geometry.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**References:**

J. Weeks, The Shape of Space.

**Fun Fact suggested by: **

Francis Su