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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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    @GerryMyerson Hmm...I never thought of that.2011-10-25

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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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    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.2011-10-25