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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  
\[2x\]
  the its volume is  
\[(2x)^3=8x^3\]
.
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  
\[2x \sqrt{3}\]

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