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What's the circumference of a circle of radius r, on a sphere of radius R?
The circumference of a circle of radius r is [itex]2\pi r[/itex], whether it is on a sphere or not.
Imagine you fix one end of a string with the length r at a point in 2d-space, and make a full circle with the other end. If you then find that the circumference of that circle is different from 2*PI*r, you conclude that the 2d-space is curved intrinsically.
That means our circumference formula becomes
[tex]2\pi r'= 2\pi R sin(\theta)= 2\pi R sin(r/R)[/itex]
That means our circumference formula becomes
[tex]2\pi r'= 2\pi R sin(\theta)= 2\pi R sin(r/R)[/itex]