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).
| . . C | . /|\ . | . |. \ . |. / | \ .
.| | \ .
. | / | \ .
. | | \ . . | / | \ .
. | 1 | r \ . . | / |h \ . . | | \ .
. | / | \ . . | | \ . . |/delta | beta \
--A------M--------H-----1-------------B------
| 1-q q |
- Let's determine some distances
h^2 + q^2 = r^2 q = (r^2)/2
h^2 + (1-q)^2 = 1 h = r*sqrt(1-(r^2)/4)
- 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')
2*beta 2*delta
F = ------ * pi * r^2 + ------- * pi - sqrt(1-(r^2)/4)
360 360
Transformed to arcus with
cos(beta) = (r^2/2)r = r/2 cos(delta) = (1-r^2/2)/1 = 1-r^2/2
F = r^2 * acos(r/2) + acos(1-r^2/2) - r * sqrt(1-r^2/4) ===========================================================
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).
Wolfgang Schumacher schumach@cs.tu-berlin.de 2B or not 2B = FF