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Despite the long title, my question is very short and clear:
Can you not just derive the surface area formula for a spherical triangle from the angle of sum formula of a spherical triangle with two simple equations?
The angle of sum formula is as follows: π+(T∆ABC/r^2)=A+B+C
And can I not just subtract pi and divide with r^2 on both sides of the equation and get the surface area formula for a spherical triangle; T∆ABC= (A+B+C- π)∙r2?
Yes, the area of a spherical triangle is proportional to the scaled angle excess. $$(A+B+C-\pi)\cdot r^2$$ is correct, although I've never before seen the area denoted with a formula symbol like $T\triangle ABC$.
I find the Wikipedia section on this to be less clear than what I'd have expected.
