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


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