Beauty of Geometry - 1

Let the radius of the bigger and smaller sphere be $\boldsymbol R$ & $\boldsymbol r$ respectively.

Now, cut the cube diagonally in 2 equal halves which goes through the center of both sphere. Then, draw a line from the center of the bigger sphere to that particular corner of the cube which will go through the center of the smaller sphere . Finally, draw two squares as shown in the picture below:

Look closely, the biggest square in the picture is the diagonal slice of the original cube. So, the other two squares which was drawn will also be diagonal slice of two other cubes. As the diagonal of a cube is equal to $\sqrt3$ times it's edge we get, $\boldsymbol {OA+AB+BC=OC}$ $\boldsymbol {\Rightarrow R+r+\sqrt3 r =\sqrt3 R}$ $\boldsymbol {\Rightarrow r=\frac{R(\sqrt3-1)}{\sqrt3+1}}$ $\boldsymbol {\Rightarrow r=R(2-\sqrt3)}$

So the radius of the smaller circle will be, $\boldsymbol {r=17(2-\sqrt3)=\color{#69047E} {\boxed{4.555}}}$