Let O be the center of the circle.

Let AB be the diameter of the circle which is one edge of the path.

Notice that the radius of the circle is  = 18 feet.


Clearly, the line AB goes through the center O.


Let CD be the other edge of the path, parallel to AB.


Draw the radius OD and notice that sin(BOD) =  = .

Hence, the angle BOD is 30°, and the angle DOC is 180° - 2*30° = 120°.


This problem is intended for an advanced student,
and the HINT that I gave in previous lines is enough for such a student to complete the solution on his/her own.