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

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

Which would lead to this:

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

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.

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

1 Answer 1


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.

  • $\begingroup$ Sorry, I forgot to include that the inner sphere would have a constant volume that would be non-zero. $\endgroup$
    – ashaw
    Feb 27, 2021 at 13:08
  • $\begingroup$ That still doesn't specify. Do you mean that both the volume between spheres and the inner sphere's volume are givens? $\endgroup$ Feb 27, 2021 at 18:44
  • $\begingroup$ Nope, the volume of the inner sphere's isn't given. If the difference between the radii is quite small, does that change things in any way? $\endgroup$
    – ashaw
    Feb 27, 2021 at 19:56
  • $\begingroup$ If you are leading up to infinitesimal, the difference in volumes is $4\pi r^2 dr.$ $\endgroup$ Feb 27, 2021 at 21:32

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