Ratio of Volume of Sphere Inside a Cube Inside a Sphere

Suppose a cube is placed inside a sphere so that the vertices just touch the sphere. Inside the cube is another sphere, with the sphere just touching the centre of each face of the cube.

If the side of the cube is  
  the the innermost sphere has radius  
The radius of the larger sphere is the distance from the centre of the cube to one of its vertices, and is equal to  
\[\sqrt{x^2+x^2+x^2} = \sqrt{3x^2} = x \sqrt{3}\]

\[\frac{Volume \: of \: Large \: Sphere}{Volume \: of \: Small \: Sphere}= \frac{4/3 \pi (x \sqrt{3})^3}{4/3 \pi x^3} = 3 \sqrt{3}\]

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