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Apologies if you've already spent a lot of time answering this one but I've spent a lot of time looking for the answer here without success unfortunately. If I bunch six spheres together to form an octahedron then how do I calculate the size of sphere that fits at the centre of those six spheres? I assume the answer to be some expression of the size of spheres containing it?

Gill Simo
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    Are the spheres all the same size? If not, it's possible that the six spheres might not be able to form an octahedron, or the size of the inner sphere might be variable depending on how the spheres are arranged. – Joseph Camacho Jul 06 '22 at 00:45

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If the spheres are all the same size, then this isn't too hard to calculate. Taking a cross section of a regular octahedron gives a square. If $R$ and $r$ are the radii of the outer and inner spheres, respectively, then $2R + 2r$ equals the diagonal of this square, $R\sqrt2$. Solving for $r$ yields $$r = (\sqrt 2 - 1) R \approx 0.414 R.$$ Picture of octahedron with spheres

Joseph Camacho
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