Chapter 33, Problem 21P

To determine

The maximum sum-of-the-angles for a triangle on a sphere.

Answer to Problem 21P

The maximum sum-of-the-angles for a triangle on a sphere is $540°$ .

Explanation of Solution

Introduction:

Sum of all internal angles of a triangle is $180°$ .

If an equilateral triangle is formed considering placing South Pole in it, then each angle will be of $60°$ , considering the surface to be flat. Now, if all three boundaries of triangle is expanded till it touches equator, then for all three lines to remain in a triangle, it is must that each angle be less than $180°$ . Now, adding up all three limiting angles of a triangle, maximum possible sum in a triangle would be $540°$ .

Conclusion:

The maximum sum-of-the-angles for a triangle on a sphere is $540°$ .