Question 80367
After the 3/4 circle at every corner the rope shortens by the length of the previous side, so the next portion of circle will not be as large. So you have 3/4 of a 50 radius circle + a sector of a 40 foot circle (blue) + a sector of a 30 foot circle (red) + a triangle with sides of 40, 22.36, and 30. minus half the area of the barn. The triangle is the overlap area of the two arcs, and its area can be figured by Heron's formula: {{{A=sqrt(s(s-a)(s-b)(s-c))}}} where {{{s=(a+b+c)/2}}}. I'll leave you to work the numbers.  You'll also need the Law of Cosines to get the angles.

Tom laidlaw
{{{drawing( 400, 400, -100, 100, -100, 100, 

rectangle(0,0,20,10), 
green( arc( 0, 0, 100, 100, 0, 270 ) ),
line(0,0,50,0),
line(0,0,0,50), 
red(arc(20,0,60,60,307,360))
blue(arc(0,10,80,80,270,342)),
red(line(20,0,38,24)),
blue(line(0,10,38,24)),
line(0,10,20,0) 
)}}}