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Measuring sum of angles of triangles on different curvature

Physics Asked on August 15, 2021

I was reading Richard Feynman’s Six Not so Easy pieces where I came across the concept of measuring geometric shapes drawn on different curved surfaces like the ones with negative and positive curvatures. So out of curiosity I drew a few triangles on a ball and using a protractor measured the sum to be approximately 210 degrees in all cases.

I wonder what would be the sum of angles of the triangle drawn on the interior of a hollow sphere. So, I took a bowl ( not an exact sphere) and drew a triangle on the inside but now I can’t measure it since the protractor won’t fit. Is there any other way to measure it? If not what would be the approximate summation? I also wonder how it would turn out on hyperboloids and more…

One Answer

Say you draw the triangle on the outside of a thin clear glass sphere. Now assume that the lines you drew were inside instead of outside. The angles would be the same, they just may be more easily measurable on the outside of the sphere.

Correct answer by Adrian Howard on August 15, 2021

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