This question is closely related to the question that I posted yesterday. The simpler symmetry of this problem makes me think that this variation will be simpler to solve, so I suggest that anyone who is intrigued by the problems should start with this one.
One way of overlapping 3 circles looks like this:
What is the smallest size that these circles can have and still be large enough to accommodate a unit circle in each region?
With unit circles pictured for clarity and zoomed in, it's something like this (where circles that appear to be approximately tangent should indeed be perfectly tangent):
EDIT (improved picture now that I know the exact radius):
By pure physical manipulation of circles, it seems to me that the radius of the 3 circles may be a bit less than 15. But I'm having difficulty determining an exact answer. Any solutions--partial or complete--would be appreciated.
Note: in writing this I realized that I have made the implicit assumption that the smallest possible area covered by 3 circles meeting these requirements will be done with 3 equally sized circles. As proof of this claim, I offer these waving hands and some mumbled lines about symmetry.
In the "perfectly tangent" configuration, we have the red Pythagoras triangle and the blue equilateral triangle, which implies the red triangle is $30^\circ, 60^\circ,90^\circ$. Therefore $$r+1 = {2\over \sqrt{3}}(r-1)$$ and $r=7+4\sqrt{3}$