G'morning David...

 DW> But the first pair of results are nothing like yours. When 
the Total
 DW> area is 36 cm^2, you calculated the Triple area to be 1.15 cm^2, and
 DW> the inter-centre separation to be equal to the radius. My program
 DW> calculates the Triple area to be 7.3 cm^2, and the inter-centre
 DW> separation to be 1.14 times the radius.

 DW> It looks like you made a slip somewhere.

Probably... it's now obvious to me that the problem-setter MUST 
have used set algebra to pose the question,  because there's no 
easy way to construct a possible answer AND THEN measure it's 
outcomes.

(Just measuring outcomes is interesting.  Ask a school student to 
draw two overlapping circles of 20cm^2 each so that the overlap 
covers exactly 9cm^2 and see how he fares.  Better still, 
determine how the marker will confirm the result ....)

The differences you mention above are those between the logic of 
the algebra (which presumes no restraint of location, no matter 
how the problem is worded) - and the logic of geometry....

Case 1 - Setting the total area to 36

Thrice coverage - set logic = 1.18, geo-prog = 7.3 and my 
eyeballing a scribble = 5ish.

Case 2 - Setting thrice to 3

Total coverage  - set logic = 50.9, geo-prog = 44.2 and my 
eyeballing the new scribble gives up.

I've had a look at all of the integral variations set logic 
permits with a total of 36 and size 20 circles,  and checked with 
enough physical rendering to show that none of them can actually 
be drawn - because of the above differences between set logic and 
geometric restraint.

However,  I can see that there will be examples that do show full 
agreement between the two modes.

For quick example, let there be 3 circles of 20cm^2 each,  
overlapping so that the area covered by all 3 is exactly 
4.48693cm^2  -  and so that the total area covered by any circle 
is 41.0268cm^2.

The area covered by any two circles alone can now be calculated by 
either set logic or geometry to arrive at a common answer - but 
only by careful contrivance.

I think we can now insist that using set logic to pose a geometric 
problem is most unlikely to produce a drawable solution.

I predict that this teacher probably copied the problem from 
somewhere else in good faith,  and that the real author worked 
from set logic using integers to create a monster that cannot 
exist on a flat surface as might be claimed.

I also predict that the original problem-setter expected the 
solution to be by way of the algebra,  especially when the 
difficulty of measuring geometric solutions is taken into account.

Phew....

:-)
 
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