The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

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   

   A Triangle with Three Right Angles   

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Geometry
High School Non-Euclidean Geometry

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.