Often, when someone introduces the idea of non-euclidean geometry they give the examples of spherical and hyperbolic geometry. To help visualize these concepts, they'll usually compare the sum of the angles of ordinary triangles in each of these geometries, as well as comparing them to a triangle in euclidean geometry.
I'm going to use the phrase "most obtuse" as meaning the triangle with the largest sum of its inner angles.
Taking the example of a unit sphere, what is the most obtuse and non-self-intersecting triangle that can be inscribed on its surface?
Best Answer
The most obtuse triangle has angles that sum to $180^\circ\cdot3-\varepsilon$ and is just shy of being a hemisphere.
Or, if you allow it, push that triangle past covering half of the sphere, then the angles are each greater that 180°. Keep pushing the triangle further and you almost cover the whole sphere, except for a small triangle and the angles sum to just under 900°