Sum of interior angle is NOT 180, or is 0 degree?

[furtivefelon]

hi, today, my teacher told us that if you draw a triangle on the surface of a sphere, you'll get a triangle with interior angle of not 180 degree, also, if you draw a triangle in a trumpet like shape, you'll get a triangle with 0 degree.. where can i find more infomation about this theory?

 

[vsage]

Your teacher is correct. The sum of the angles on the surface of a sphere is > 180, but IIRC is < 180 on say the surface of a saddle. The only real word phrase I can give you to use in your search for these kind of counterintuitive situations is "noneuclidean geometry", or "spherical geometry". I'll try to fish up a good link.

 

[Tide]

Pop quiz

What's the maximum value of the sum of the interior angles of a triangle on the surface of a sphere?

 

[Gokul43201]

Who're you asking ?

 

[Gokul43201]

On a sphere, a great circle is a ...

 

[Math Is Hard]

Ouch! Gokul, you're hurting my brain! What's a "great circle"? I am dying to know... just curious... I can't yet discern differences of a circle drawn on a sphere as opposed to a circle drawn on a flat sheet of paper.. except that a circle drawn on a sphere would be drawn around a slight bulge.

 

[Math Is Hard]

I hope that's not too dumb a question .. I'd just really like to know.

 

[Tide]

Who're you asking ?

Anyone who cares to jump in!

 

[vsage]

I mentally constructed a triangle whose interior angles summed to 540 degrees in my head. I say in my head because I thought of it but there's no telling whether the mental construction actually summed right in my head. I give up if it's any more than that. My head hurts now

 

[Galileo]

I think you can get the sum of the angles as close to 540 degrees as you like.
Just a guess.

 

[Tide]

540 is a bit light - sorry, Vsage! :-)

 

[Integral]

Consider the intersection of 2 great circles.

A great circle is the circle that is formed by intersection of the surface of a sphere and a plane that passed through the center of the sphere. For example the equator.

 

[jcsd]

What's the maximum value of the sum of the interior angles of a triangle on the surface of a sphere?

I'll take a wild guess: 3*360 - 180 (i.e. 900 degrees). On a sphereyou say what would appe appear to be the exterior angle sare unf act the inerior angles and the minimum value of the sum of the inetrior angles is 180 degrees in the degnerate case (not that it really makes sense to talk about the sum of the interior angles in the degenerate case).

 

[Galileo]

540 is a bit light - sorry, Vsage! :-)

How can it be larger than 540 degrees? When measuring an angle between lines we have two choices and we always take the smaller one.
When an angle between lines gets larger than 180, we take the other angle, which is smaller than 180 degrees. So the maximum angle would be 3*180=540...
Although in this case it could mean measuring the angle outside the triangle, which is not the idea.
Hmm... on second thought, it might be possible to have an angle greater than 180 degrees, but you'd have a weird way of looking at the triangle. The sides through that point where the angles is greater than 180 degrees have to meet on the sphere and you'd be measuring the angles on the outside of the triangle, while you could just as well measure them on the inside...

 

[Tide]

Galileo,

Jcsd's wild guess is a very good one!

For arbitrarily small triangles on a spherical surface the sum of the interior angles approaches 180 degrees. However, a spherical surface has the interesting property that the remaining surface is also a triangle! An interior angle of the small triangle and the corresponding interior angle of the large triangle sum to 360 degrees.