Human hamster ball

Geometry Level 5

A zorb is a large sphere made out of transparent plastic, in which a rider can roll downhill.

A certain zorb has a circumference of \(9 \text{ m}\). Within this special zorb, two layers of fabric are present to protect the rider. The outer one is shaped into a cube, with each of its vertices attached to the edge of the outside, spherical layer. Like the outermost layer, the innermost one is also shaped into a sphere, with its fabric attached tangentially to each of the cube's six faces.

If the volume of the inner sphere can be expressed in the form \[\frac { A\sqrt { B } }{ C{ \pi }^{ C } } {\text{ m} }^{ 3 }\]

where \(A,B\) and \(C\) are integers with \(B\) square-free, find \(A+B+C\).

This problem is original.

Picture credits: My First Zorbing by Josh Berglund, Wikipedia


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