Ratio of Volume of Cube Inside a Sphere Inside a Cube

Suppose a cube is placed inside a sphere so that the vertices of the cube just touch the sphere. The sphere is placed inside a cube so the sphere just touches the centre of each face of the cube.

If the inner cube has side  
  the its volume is  
The distance from the centre of the innermost cube to a vertex of the cube is equal to the radius of the circle and is  
\[\sqrt{x^2+x^2+x^2} = x \sqrt{3}\]
The side of the large cube is twice the radius of the sphere, and is equal to  
\[2x \sqrt{3}\]

\[\frac{Volume \: of \: Large \: Cube}{Volume \: of \: Small \: Cube}= \frac{(2x \sqrt{3})^3}{8x^3} = 3 \sqrt{3}\]

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