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.
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.
