The Gemara Eiruvin 56b said that the Migrash (the orange area) is one quarter of the area of the sados (the green area). Let's look at the math. (This is how I understood the maskana)
greenArea = (2r+2000+2000)^2 - orangeArea - cityArea(since the length of the migrash is 1000 amos + the length of the techum is 1000)
orangeArea = (2r+1000+1000)^2 - cityArea
cityArea = 3r^2
greenArea = (2r+4000)^2 - (2r+2000)^2 - 3r^2
orangeArea = (2r+2000)^2 - 3r^2
greenArea/orangeArea = ((2r+4000)^2 - (2r+2000)^2 - 3r^2 + 3r^2)/((2r+2000)^2 - 3r^2)
greenArea/orangeArea = ((2r+4000)^2 - (2r+2000)^2)/((2r+2000)^2 - 3r^2)
According to Raba, this works if r = 1000 (the city is 2000 x 2000).
However, the result will be 20/13? Where did I go wrong?
I saw the Soncino Gemara and they explain that the Migrash is also a circle. In that case, we have the following picture:
BigArea = (2(r+1000+1000))^2
CompleteMigrashArea = 3(r+1000)^2
City = 3r^2
GreenArea = BigArea-CompleteMigrashArea
migrashArea = CompleteMigrashArea-City
GreenArea = BigArea-(MigrashArea+City)
GreenArea = BigArea-MigrashArea-City
GreenArea = (2(r+1000+1000))^2-3(r+1000)^2-3r^2
MigrashArea = 3(r+1000)^2 - 3r^2
GreenArea/MigrashArea = ((2(r+1000+1000))^2-3(r+1000)^2-3r^2)/(3(r+1000)^2 - 3r^2)
If r = 1000, then then the ratio is 7/4
Doesn't work either.
Abaye says the same method will work if the city is 1000 by 1000. (r=500).