Do you know the diameter of the core and the radius of the sphere? That should be quite enough to decide:
1) How much of the sphere to subtract because you cut it away, and
2) How much of the core to add because you exposed it.
I was wondering, how would you calculate the surface area of a sphere with a "cylinder" (I say "cylinder" rather than cylinder because the cylinder would have curved ends rather than flat) cut into it. The example I would use is that if you had a perfectly spherical apple and then cored it, how would you calculate the surface area of the cored apple?
Well, it takes a little work for the subtraction of the spherical caps, but the addition of the cyllindrical core is a right triangle probem.
Considering only the top half the sphere (you can double it later), the height of the cyllinder is . Thus, the lateral surface area of the top half of the cyllinder is simply:
Now, you do the spherical caps to subtract. Come on, I expect the ancient Greeks could do it.
Well, I don't know what any of that means, but you'll need two things for the cyllinder.
1) Pythagorean Theorem. Hypotenuse is r1. One leg is r2. You must solve for the other leg. The solution is as I have posted it: h = ??
2) Formula for the lateral surface area of a right circular cyllinder.
The surface area of one cap appears to be . You should check that out and make sure it looks reasonable. I'm sure there is some lovely geometry way to discern this, but I haven't managed to think of it this evening.
It is not that hard a problem with a little calculus. This may be part of the reason why I failed to motivate myself to find the geometry solution.