Let "x" be the unknown side of the lawn. A square lawn surrounded on three sides by the path 4 m wide is a rectangle of dimensions (x + 2*4) = (x+8) and (x+4) meters. The area of this rectangle is (x+8)*(x+4) m^2. Then the area of the path is= = 12x + 32 m^2. The condition says that 12x + 32 = . It is = 0. Solve this quadratic equation for x. Answer. x = 16 m.