This question is closely related to a question that I posted recently.
Another way of overlapping 3 circles looks like this:
If we want to size these circles such that they are as small as possible yet still large enough for a unit circle to fit in each of the 7 regions, then we get a diagram like this:
The question then, is what are the radii of the circles? By convention, let's call the larger circle A and the 2 smaller circles will be B. (Or B1 and B2 if you had any desire to differentiate them.)
I can offer a variation on this diagram with a large number of colored triangles drawn on it... but I cannot at this moment offer one with useful triangles drawn on it. Likewise, I can offer the guidance that A appears to have a radius that is a smidge larger than 5 while B's seems to be just a tad more than 4. But I haven't been able to figure out an exact answer.
Let the big circle radius be $R$ and the small two circle radius $r$.
As shown in picture, we have the red Pythagoras triangle implying $$(r+1)^2=(r-1)^2+(R-1)^2$$
The blue line segment calculation shows $$r=R-(r-1)+2$$
The second equation simplifies to $2r=R+3$. Solve the first we get $4r=(R-1)^2$ so $2R+6=(R-1)^2$ and $R=5$ as we take the positive solution. $r=4$ follows by substituting in to the other equation.