# Internal angles of a triangle

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 592168.  Thu Jul 30, 2009 8:50 am Hi everyone. This is my very first post on these forums, although I sincerely hope it won't be my last. :) I really want to see how many of you know the answer to this question: What do the internal angles of a triangle always add up to?

 592178.  Thu Jul 30, 2009 8:57 am That depends what you mean by triangle.

 592188.  Thu Jul 30, 2009 9:03 am A geometric shape with 3 distinct, straight sides.

 592195.  Thu Jul 30, 2009 9:12 am Welcome to the forums Amadeus! I'll give the answer less than 180 degrees. I've assumed that you are talking about in hyperbolic geometry, yes? ;)

 592202.  Thu Jul 30, 2009 9:17 am Well you could equally say greater than 180 degrees, if the triangle is on a convex surface. And welcome to Qi Amadeus.

592203.  Thu Jul 30, 2009 9:18 am

 Davini994 wrote: Welcome to the forums Amadeus! I'll give the answer less than 180 degrees. I've assumed that you are talking about in hyperbolic geometry, yes? ;)

Thanks Davini!

I'll let this question simmer for a little bit before giving an explicit solution.

 592208.  Thu Jul 30, 2009 9:27 am I prefer to say “π radians” (Euclidean geometry), “less than π radians” (hyperbolic geometry) or “more than π radians” (elliptic geometry).

 592233.  Thu Jul 30, 2009 9:56 am 42

 592245.  Thu Jul 30, 2009 10:05 am (I just posted the answer but I just thought that maybe I should wait if more people want to come and have a go at the puzzle. What do you think?)

 592261.  Thu Jul 30, 2009 10:32 am Welcome, Amadeus :-)

 592282.  Thu Jul 30, 2009 11:49 am Welcome :)

 592293.  Thu Jul 30, 2009 12:17 pm I'm with Susannah; degrees are an unnatural heresy not worthy of inclusion in serious mathematics. Still, at least they're better than grads. Oh, and I'd add that the sum will in all cases be at least 0 and at most 2π.

 592297.  Thu Jul 30, 2009 12:21 pm Not sure on your at most 2π Colin. Try drawing a triangle on the surface of a sphere, it's pretty easy to get one with interior angles greater than 2π.

 592305.  Thu Jul 30, 2009 12:52 pm Sorry, yes. I guess the upper bound would be 6π.

592516.  Thu Jul 30, 2009 5:20 pm

 ColinM wrote: Sorry, yes. I guess the upper bound would be 6π.
I can imagine special cases where there would be no upper bound - but the geometry would be potentially discontinuous and/or require four dimensions.

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