Goats never eat all the grass, leaving the roots intacts, so you can just let the goat roam free in the enclosure. It'll eat roughly the upper half of each blade of grass.
Ok, trick answers aside, it still seems straightforward enough.
Let r1 = 100m, r2 = length of the rope.
Our points are
C: center of the paddock
T: tying point on the fence
I1, I2: the two intersections between the circles (C, r1) and (T, r2)
alpha: angle I1, C, I2
beta: angle I1, T, I2
If you want to graze exactly half the grass, then C, I1, T, I2 is convex, so alpha < 180°, beta < 180°. That means that the grazed part is the sum of two circular segments
, with a known formula for area:
A = r1^2/2 (alpha - sin(alpha)) + r2^2/2 (beta - sin(beta))
with A being the area of the grazed part, which is half of the paddock's area:
A = 1/2 * pi * r1^2
We have an equation with three variables r2, alpha, beta. The fourth, r1, is a constant.
To get rid of two of the variables, look at the triangle C, I1, T. It has two sides with length r1 and one with r2, one angle (alpha/2) and two angles (beta/2). It's an isosceles triangle
, so you can find formulas for the relations between alpha, beta and r2. Substitute in the equation above, solve for alpha, beta or r2, whatever seems easiest. To solve for alpha, you could use:
beta/2 = (180° - alpha/2) / 2
r2 = 2*r1* sin(alpha/4)
Solving for anything is tricky, because the formulas are a tangled mess of triginometric functions that are difficult to untangle. At this point, I'd just solve numerically.