Tangent sphere to a cube

Geometry Level 5

A sphere of radius \(3\sqrt{2}\) is tangent to the edges \(AB,\) \(AD,\) \(AA_1,\) and the face diagonal \(CD_1\) of the cube \(ABCDA_1B_1C_1D_1\).

The volume of the cube can be written as \(a+b\sqrt{c}\), where \(a,\) \(b\) are integers and \(c\) is a square-free positive integer. What is the value of \(a+b+c\)?

Details and assumptions

The order of the vertices in the cube is shown on the picture below.

Face diagonal simply means the diagonal of a face. It is technically not an edge of the cube, which is why we denoted \(CD_1\) separately.


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