From the condition, the slant height  R is 4 times the cone radius  r :

    R = 4r.              (1)


If  is the angle under the question, then the length of the arc of the sector is

    s =  = .    (2)

   
At the same time, this length "s" is the circumference of the base of the cone:

    s = .           (3)


Equating (2) and (3), you get

     = .


Canceling "r" in both sides, you get

     =  = .       ANSWER


Answer.  The angle under the question is  .