# Calculating radii (or difference of radii) of concentric spheres with spherical shell volume

I'd like to calculate the radii or the difference of the radii (thickness of spherical shell) of 2 concentric spheres where the only given value is the volume between the concentric spheres (spherical shell). The inner sphere has a constant volume (non-zero).

I know the volume of the spherical shell's equation would be like this:

$$V = 4/3\pi(R^3-r^3)$$

I thought this formula might be involved for the cubic differences of the radii

$$(x^3-y^3) = (x-y)(x^2+xy+y^2)$$

$$V = 4/3\pi(R-r)(R^2+Rr+r^2)$$

I'm not sure where to go from here. I would also be happy to know if there are any alternative methods for the solution.

• Are you looking for the napkin ring problem? The volume in this problem does not depend on the sphere's radius. Feb 27, 2021 at 2:12
• @TobyMak It sounds similar, but how I'm not sure how I could apply that here Feb 27, 2021 at 19:59

Your question is underdetermined. For a given volume, we could simply have the outer sphere have that volume and the inner sphere have volume $$0$$. Or we could move the outer sphere's surface outward and move the inner sphere's surface outward a somewhat greater amount to keep the volume between them constant.
EDIT: For example, suppose the difference in volumes of the two spheres is $$\frac{4\pi}{3}\cdot 16^3$$. This can be accomplished with radii $$0$$ and $$4$$, or with any inner radius you want, say $$5$$, you can calculate what the outer radius must be:$$\frac{4\pi}{3}R^3-\frac{4\pi}{3}5^3 = \frac{4\pi}{3}\cdot 16^3$$ Which you can solve for $$R$$. Replace $$5$$ and $$16$$ by any values you like and solve for the outer radius needed.
• If you are leading up to infinitesimal, the difference in volumes is $4\pi r^2 dr.$ Feb 27, 2021 at 21:32