Recently my differential geometry lecturer demonstrated that the sum of the interior angles of a triangle in a sphere is not necessarily never $180^\circ$. This is one way to prove that the earth is not flat. I was wondering, what then is the maximum sum of the interior angles of triangles in a sphere, since this sum is not a constant?
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8$\begingroup$ The maximal sum of interior angles is achieved by drawing a very small triangle somewhere on the sphere and then declaring the inside to be the outside and vice versa. The sum of the interior and exterior angles is necessarily always $3\times 360^\circ$ and since one of these sets cannot sum to less than $180^\circ$, the opposite one cannot be more than $5\times 180^\circ$. $\endgroup$– hmakholm left over MonicaCommented Oct 24, 2011 at 23:52
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3$\begingroup$ The "not necessarily" in the question should be replaced by "never" (unless one is prepared to accept degenerate triangles, where all three vertices lie on a line). $\endgroup$– Gerry MyersonCommented Oct 25, 2011 at 0:04
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1$\begingroup$ @GerryMyerson Hmm...I never thought of that. $\endgroup$– Samuel TanCommented Oct 25, 2011 at 0:25
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Perhaps your teacher taught you something like this from Wikipedia: $$180^{\circ}\times\left(1+4 \tfrac{\text{Area of triangle}}{\text{Surface area of the sphere}}\right)$$
If you are prepared to have a triangle which has more than half the area of the sphere then the maximum can approach $900^\circ$; if not then $540^\circ$.
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1$\begingroup$ "Maximum can approach"? Does that mean the max doesn't exist, and I should be asking for the supremum instead? That seems to tie in with that Henning said earlier. $\endgroup$ Commented Oct 25, 2011 at 0:27
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1$\begingroup$ You get an angle sum close to $180^{\circ}$ with a very small triangle. Turn this inside out and you get an angle sum close to $900^{\circ}$ with a triangle which covers almost all of the sphere. In each case the limit is a degenerate triangle (as Gerry Myerson said) and so perhaps cannot be achieved. $\endgroup$– HenryCommented Oct 25, 2011 at 11:17