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            I want to calculate the white area enclosed by four circles (as indicated by the arrow). Spacing between the center of circles 'd' and radius is 'r'.
           x1^2+y1^2=r^2 and also that (x−d)^2+(y−d)^2=r^2

overlapping circles

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  • $\begingroup$ What are the equations of the two other circles? $\endgroup$
    – cvgmt
    Jun 9, 2021 at 2:18

2 Answers 2

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shift = 3/2 ;

c1 = ImplicitRegion[(x + 1/2*shift)^2 + (y + 1/2*shift)^2 <= 1, {x, y}] ;
c2 = ImplicitRegion[(x - 1/2*shift)^2 + (y + 1/2*shift)^2 <= 1, {x, y}] ;
c3 = ImplicitRegion[(x + 1/2*shift)^2 + (y - 1/2*shift)^2 <= 1, {x, y}] ;
c4 = ImplicitRegion[(x - 1/2*shift)^2 + (y - 1/2*shift)^2 <= 1, {x, y}] ;

r1 = RegionUnion[c1, c2, c3, c4] ;
r2 = Rectangle[{-1/2*shift, -1/2*shift}, {1/2*shift, 1/2*shift}] ;

RegionPlot[{r1, r2}]
Area[r2] - Area[RegionIntersection[r1, r2]]
(* 9/4+1/4 (-3 Sqrt[7]-8 ArcSin[3/4]+8 ArcSin[Sqrt[7]/4]) *)

enter image description here

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  • $\begingroup$ How about shift=1/4? shift=5? $\endgroup$
    – user64494
    Jun 9, 2021 at 7:03
  • $\begingroup$ @I.M. it working for me. $\endgroup$ Jun 9, 2021 at 8:34
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One should not use a computer if the calculation can be done easily with paper and pencil.

Consider a circle segment:

enter image description here

the area of this circle segment is:

alpha = 2 ArcCos[d/2/r];
circsegarea= r^2/2 (alpha-Sin[alpha]);

Now if we subtract from the square d^2 a full circle we subtract too much, namely 4 circle segments. Therefore, the searched for area is:

area[d_, r_] = d^2 -  r^2 Pi + 4 circsegarea

The example with r=1 and d=1.5 gives:

area[1.5, 1]
(* 0.0150309 *)
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