What is the average distance between points in a circle that lies on a sphere?

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For a sphere of radius R with a circle of intrinsic radius r on it, what would the average intrinsic distance be between points in that circle? I'm as interested in approaches/methods as I am in the actual solution. Is there a way to do something like that for n-balls on n-spheres? Any insights in how this would generalise to hyperbolic geometry also welcome.