text: you have a round field.a goat is tethered at one end to eat exactly half of field what is lenth of rope?   agree with hayharbr, but in this case, we can use   1) the relation between central angle and inscribed angle (central angle = 2 * inscribed angle)  2) the trigonmetric formual for the area of a triangle (s(abc) = [b*c*sin(a)]/2)  equation i: area of the crescent = half area of the field [r1^2 * alpha/2] - [r2^2 * alpha/4] - [r1^2 * sin(alpha)]/2 + [r2^2 * sin(alpha/2)]/2 = pi/2 * r1^2  equation ii:  r1 * sin(alpha/2) = r2 * sin(alpha/4)  where r1 = radius of the field r2 = length of the rope  above system of 2 equations is sovable for r2 against r1. as hayharbr said, it can take a long time to solve manually!  further comments:  1) if things do not simplify, we shall get a solution with reverse trigonometric function (arcsin).  2) what about the approximation  r2 = sqrt[r1^2 + (r1^2)/4 + (r1^2)/16] r1 = radius of the field r2 = length of the rope  3) hope you allow considering that the field is flat, neglecting also the sphericity of the earth. if not, i surrender.  4) finally, we are lucky to have a round field. think how is the shape of the fields of our neighbors and how they should solve the same problem for them.