G'morning David, 

 DW> Diagrams that look like the one in my stepson's math homework are used
 DW> in various branches of math.

Some years ago,  our Dept of Education went barmy over "new maths" 
when some key teacher re-discovered the world of sets, groups and 
mappings that the Rev's Dobson and Venn used to delight in...

 DW> However, the question he had to answer was
 DW> a simple geometrical one, with no underlying significance. It dealt
 DW> only with overlapping circles and enclosed areas.

 -> Ergo,  the `total area enclosed by exactly two circles'
 -> is 18 (cm^2)

 DW> I agree that the answer is 18 cm^2, *if* there is an answer.

You'll recall my reluctance to dig out my drawing set...I once 
tried to write a Venn diagramme drawing programme,  giving it away 
when I ran into this kind of hassle...

 DW> I have proved that the symmetrical case, in which the centres of the
 DW> circles are at the vertices of an equilateral triangle, is NOT
 DW> possible.

Well, I've succumbed and determined the radius of one circle to be 
2.5231325 odd cms before calculating the angle of a chord whose 
area was 4.5cms^2  -  half of the overlap of 9cms^2 needed between 
any two squares  -  at 126.84910 degrees or so.

Exhuming ancient instruments (originally my grandfather's), I drew 
me a 20cm^2 circle with radius 2.52cms,  constructed two radii at 
126.85 degrees and drew in the chord.  It looked good...

Next, I used my compasses to locate the centre of the second 
circle from the bases of the chord,  and drew that in.  Still 
looked good - about 9cms in the overlap,  and about 11cms on each 
side.

My equilateral now had sides of about 2.3cms,  and the 3rd circle 
fitted physically.

However, writing in the logical cm^2's of each segment shows 
clearly a very bad fit with the problem as set.

The outer single-coverage segments of `5cms' are obviously much 
larger than the double-coverage `6's - which are individually 
smaller than the central `3' of treble coverage.

I cannot visualise a 3d surface which might improve the fit, and, 
although some interesting eclipse might satisfy the specification,  
I'm not about to go hunting for it - such a shape fails the 
requirement of being a circle in the original question...

The solution ought to be symetrical to satisfy the logic - so I'd 
also rule out attempting interesting asymetricities.

I suspect the problem was written under the misapprehension that 
group or set algebra can be mapped geometrical  -  has the setter 
of the problems produced a model answer ?

A challenge by way of wager could be profitable ?

:-)

--- Maximus/2 3.01