The normal surface area of a sphere is \(4\pi R^2\)but this is for a Euclidian 3 space. However around a gravitating body of mass M space-time has curvature. The metric of space becomes progressively more disturbed closer to such an object. Light cones are tipped more and more with proximity along radial lines. Tangential shortest paths become curves.

Because of curvature the volume of the sun is about 6 earth volumes larger than the Euclidian volume equation would predict.

I'm aware of the Schwarzschild metric and one could integrate it along a path of constant R to determine the proper circumference. But I'm not sure how that should be used to evaluate the surface area at radius R, maybe this is just a double integral over azimuth and elevation (with a coordinate system correction)?

In addition, the calculation of the surface area of the event horizon of a black hole seems to be repeatedly presented as though it is consistent with using the standard formula for spherical surface area which makes no sense to me, especially for small black holes where there are expected to be huge tidal forces just outside the event horizon.

In addition, should the radius / circumference / area / volume for a large star or black hole be based on proper distances or distances from the point of view of an observer at infinity, from the perspective of calculating other physics such as internal processes, thermodynamics, entropy, or black body radiation?