# Calculating the area of star shaped region by four partially overlapping circles

            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


• What are the equations of the two other circles? Jun 9, 2021 at 2:18

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]) *)


• How about shift=1/4? shift=5? Jun 9, 2021 at 7:03
• @I.M. it working for me. Jun 9, 2021 at 8:34

One should not use a computer if the calculation can be done easily with paper and pencil.

Consider a circle segment:

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 *)