The pasture is the circular area around M with the radius of 1. The goat is nailed on to B and has a length of r. These circles intersect in C resp C' (mirrowed to the x-axis).
- Let's determine some distances
- The overlapping area of the two circles is composed of
the circle segment BCC' plus the circle segment MCC' minus the 2 triangles MBC and MBC'(double counted by BCC' and MCC')
Transformed to arcus with
cos(beta) = (r^2/2)r = r/2 cos(delta) = (1-r^2/2)/1 = 1-r^2/2
Now you can determine the required radius by a numerical approximation. For F = pi/2 the goat's length computes to 1.15872847302 (meters, miles or feet).