3-D geometry question 
I tried to do draw the problem in 3D but I stil can't imagine how such sphere can exist... it's really confusing

Let us first find out which spheres "fit snugly" against the arc between $A$ and $B$.
Such a sphere must obviously fit against the whole circle of which the arc is a part.
This means that the center of the sphere must be on the symmetry axis of the circle (the line normal to the circle's plane that goes through its center).
If $D$ is placed at the origin (and other points are treated as vectors), this line goes through the point $A+B$ in the direction $C$.
Similar considerations of course can be made for the other two arcs.
We obtain three lines, and the center of the sphere should be on all of them (because the sphere should fit against all arcs).
Fortunately these lines do intersect, and this happens at the point $A+B+C$.
This is the opposite point of $D$ in the cube which has the edges $DA$, $DB$ and $DC$.
The radius you need is now the distance from this point to any of the points $A,B,C$.
It should now be simple enough to finally figure out the radius.
(Do ask if clarification is needed.)
