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Calculate the area of the spherical triangle defined by the points (0, 0, 1), (0, 1, 0) and (1/√2, 0, 1/√2)

Own work:

From the spherical Gauss-Bonnet Formula, where T is a triangle with interior angles α, β, γ. Then the area of the triangle T is α+β+γ-π.

How do I work out the interior angles in order to use this formula?

Any help appreciated.

1 Answers 1

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Use dot products of the coordinate vectors to show that two sides of the triangle measure $90°$ of arc and the the third side, opposite $(0,1,0)$, measures $45°$. It's as if $(0,1,0)$ were the North Pole and the other two vertices are $45°$ apart on the Equator, so the triangle covers one eighth of the Northern Hemisphere. You should see the area clearly now, without need for the full Gauss-Bonnet formula.

  • 0
    Can you please explain this further as I am struggling to picture it and form an answer – 2017-02-25
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    I am not good with pictures, can someone put one in with the specifications given? – 2017-02-25
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    If not can you show me some workings out as a head start so I can see how to approach it – 2017-02-25