I would've thought the bread you eat ends up looking like this, in which case the answer is 1/4.
But that only considers distance in the direction to the origin, not how the sides of the square are closer than the corners are, so the corners need to be rounded off too.
Edit, here's my attempt with Cartesian coordinates rather than Polars. Starting with the upper right quadrant as before, calculating the lower half, thus an eighth of total area. For a given (x,y), we have the distance to the origin being r=(x
2+y
2)
1/2, and the distance to the right hand edge being (1-x). Thus solving for r=1-x gives y=(1-2x)
1/2. This is valid only until the top edge becomes closer, IE until y=x. Therefore this intersects with the line y=x at x=(1-2x)
1/2, x=-1 +- 2
1/2, so that's where this eighth of the total area ends. Call this location L.
We therefore have the area just being under the y=x curve until point x=L, where it then turns into the weird curve y=(1-2x)
1/2 until x=1/2. Thus we have total area
A= Int{x, x=0, L} + Int{(1-2x)
1/2,x=L,1/2}
= L
2/2 + ( -1/3.(1-2x)
3/2 , x=L, 1/2 )
=(3-2.2
1/2)/2 + (0 + 1/3.(3-2.2
1/2)
3/2)
=(3-2.2
1/2)/2 + (5.2
1/2-7)/3
=(9-6.2
1/2 )/6 + (10.2
1/2-14 )/6
=(-5 + 4.2
1/2 )/6 =~ 0.11.
Thus total area 8A~=0.88, thus ratio 8A/4=2A=0.218951416, as before.
...no.
Still not finished, definitely need an extension. This stupid level is being a huge roadblock, not finding any viable solutions. Don't expect it finished this year.