A question about spherical geometry

[qfxz]

What is the range of the sum of a sperical triangle's interior angles? Is it (π, 2π) or (π, 3π)? My high school textbook says it is (π, 2π), but some websites say it is (π, 3π).

 

[qfxz]

I mean spherical triangle. Sorry for mistyping.

 

[Phrzby Phil]

Have you tried constructing one? Try vertex at the North Pole almost pi, other 2 vertexes on the equator.

 

[MathLover1]

The sum of angles of a triangle formed in a spherical geometry is equal to 180° , so 2π

 

[Phrzby Phil]

If the North Pole angle is pi/2, and the base angles are on the equator, then they are each also pi/2, then this triangle's angles add to 3pi/2.

Widen the North Pole and you're even greater.

 

[MathLover1]

In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. On a sphere, however, the corresponding sum is always greater than 180° but also less than 540°.

 

[nycmathguy]

In a small triangle on the face of the earth, the sum of the angles is only slightly more than 180 degrees. On a sphere, however, the corresponding sum is always greater than 180° but also less than 540°.

Can you please show a graph to represent what you are saying here?

 

[MathLover1]

 

[nycmathguy]

1. How did you make this drawing?

2. What do the letter mean? Angles? Points?

 

[MathLover1]

I found image
The diagram shows the spherical triangle with vertices A, B, and C. The angles at each vertex are denoted with Greek letters α, β, and γ. The arcs forming the sides of the triangle are labeled by the lower-case form of the letter labeling the opposite vertex.

 

[nycmathguy]

I found image
The diagram shows the spherical triangle with vertices A, B, and C. The angles at each vertex are denoted with Greek letters α, β, and γ. The arcs forming the sides of the triangle are labeled by the lower-case form of the letter labeling the opposite vertex.

This looks complicated. I will stick to high school geometry.