Sum of Angles of a Triangle in Non-Euclidean Geometry

Date: 08/01/2001 at 18:54:21
From: Roger
Subject: Three-dimensional figures

Recently I was trying to teach my son that the sum of all the angles 
of a triangle is equal to 180 degrees. My son came up with a drawing 
of a triangle on a surface of a sphere in which the sides, due to the 
shape of the sphere, looked like an arch instead of a straight line 
of a normal triangle drawn on a plane surface. When we added the 
angles of that particular triangle the sum was definitely greater than 
180 degrees. Is that possible? Why?

Date: 08/01/2001 at 23:11:47
From: Doctor Peterson
Subject: Re: Three-dimensional figures

Hi, Roger.

The theorem about the sum of angles in a triangle is from PLANE 
geometry, and depends on the parallel postulate, which says that, 
given a line and a point, there is exactly one line through the point 
parallel to the line. If you move to SPHERICAL geometry, this 
postulate is no longer true, and you have a "non-Euclidean" geometry. 
(You have to redefine "lines" to be great circles.) Here, the sum of 
angles is always greater than 180 degrees; in fact, it turns out that 
the excess over 180 degrees is proportional to the area of the 
triangle. Try making a triangle containing 1/8 of a sphere, using the 
equator and the 0 and 90 degree longitude lines; then try some other 
similar triangles to see that this is true.

You can read about this in the Dr. Math archives:

   Drawing Triangles
   http://mathforum.org/dr.math/problems/joe6.18.97.html   

   A Triangle with Three Right Angles
   http://mathforum.org/dr.math/problems/stine.12.1.99.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/