->  -> Cover  Segment Cm^2  Total Cm^2 
 
->  -> Thrice  1.1572999     1.1572999 
->  -> Twice   7.2284667    21.6854002 
->  -> Once    4.3857666    13.1572999 
->  ->         Totalling... 36.0000000 
 
-> Here the optimum area covered by all three circles is 1.16cm^2,  
-> NOT 3 as set.  Worst - when I draw it, the equilateral triangle  
-> has sides equal to the radii,  and the centre section is obviously  
-> close to 5cm^2  - NOTHING like 1,16.... 
 
-> Next, setting the thrice to 3 cm^2 and leaving the total area  
-> unconstrained then has that total coverage rise.... 
 
->  -> Thrice  3             3.0 
->  -> Twice   1.0237947     3.0713841 
->  -> Once   14.952411     44.857233 
->  ->         Totalling... 50.9286171 
  
I fiddled with my computer program so as to make it calculate the 
distance between the centres of the circles (ASSUMING them to be at the 
vertices of an equilateral triangle) in the two cases when the "total" 
area is 36 cm^2 and when the "triple" area is 3 cm^2. It also 
calculated and printed both areas and the inter-centre separation in 
both cases. 
  
When the Total area is 36 cm^2, the inter-centre separation comes out 
to be 0.77 times the radius of each circle, and the Triple area is 7.3 
cm^2. 
  
When the Triple area is 3 cm^2, the inter-centre distance is 1.14 times 
the radius, and the Total area is 44.2 cm^2. 
  
My second pair of results is pretty much the same as yours. The small 
differences are probably due to quantization approximations in my 
program. 
  
But the first pair of results are nothing like yours. When the Total 
area is 36 cm^2, you calculated the Triple area to be 1.15 cm^2, and 
the inter-centre separation to be equal to the radius. My program 
calculates the Triple area to be 7.3 cm^2, and the inter-centre 
separation to be 1.14 times the radius. 
  
It looks like you made a slip somewhere. 
  
I'll append the program below. 
  
                               dow 
  
------------------------------------------------------- 
  
DEFDBL A-Y 
DEFLNG Z 
CLS 
PI = 4 * ATN(1) 
K = 9 * PI / 20 
L = .2 
H = 1 
DO 
D = (H + L) / 2 
 IF ABS(H - L) < .005 THEN EXIT DO 
 X1 = -D: Y1 = 0 
 X2 = D: Y2 = 0 
 X3 = 0: Y3 = D * SQR(3) 
 Z1 = 0: Z2 = 0: Z3 = 0 
 FOR X = X1 - 1 TO X1 + 1 STEP .01 
  FOR Y = -1 TO 1 STEP .01 
   DX = X - X1: DY = Y - Y1 
   IF DX * DX + DY * DY < 1 THEN 
    Z1 = Z1 + 1 
    DX = X - X2: DY = Y - Y2 
    IF DX * DX + DY * DY < 1 THEN 
     Z2 = Z2 + 1 
     DX = X - X3: DY = Y - Y3 
     IF DX * DX + DY * DY < 1 THEN 
      Z3 = Z3 + 1 
     END IF 
    END IF 
   END IF 
  NEXT 
 NEXT 
 Z4 = Z2 - Z3 
 Z5 = Z1 - (2 * Z2 - Z3) 
 ' execute one or other of following 2 lines 
 IF Z3 < 3 / 20 * Z1 THEN H = D ELSE L = D ' triple area = 3 
 'IF Z3 + 3 * (Z4 + Z5) > 36 / 20 * Z1 THEN H = D ELSE L = D'total=36 
LOOP 
F = 20 / Z1 
PRINT "Separation of centres:"; 2 * D; "x radius" 
PRINT "Triple overlap area:"; F * Z3 
PRINT "Total area covered:"; F * (Z3 + 3 * (Z4 + Z5)) 
  
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