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Consider the unit sphere in $R^3$ and an equilateral triangle of side length 1, with all vertices on the surface of the sphere. Now project (from the center of the sphere outward) the triangle and its circumscribed circle onto the surface of the sphere to get a spherical triangle and its circumscribed spherical circle (the area covered by which would be dome-shaped).

I am trying to find the relation between the area of the spherical triangle and the area of the circumscribed spherical circle, not only in $R^3$, but in general. That is, in $R^{n+1}$ with $S^n$ and an $n$-simplex of unit side length with all vertices on the surface of $S^n$ projected onto the surface of $S^n$ along with its circumscribed $(n-1)$-sphere.

This is because I think it will be easier to compute the area of the spherical disk (equivalently, area covered by the spherical circle) rather than the area of the spherical triangle, which is my ultimate goal. Ideally, this could be generalized even more to a sphere of radius $r$ and a simplex of side length $r$, but right now I'm just trying to get a foothold of this problem.

Any help in this direction is appreciated!

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