When we flatten out a cone into the sector of a circle,
the radius of this circle is the slant height of the cone,
while the arc of this sector is the circumference of the base circle of the cone.


So, we should find the slant height and the circumference of the base circle.


The slant height is the hypotenuse of the right triangle, whose legs are
the radius of the base circle and the height of the cone.

So, in our case, the slant height is  

    h =  =  =  =  = 13 cm.


The circumference of the base circle is

    C =  =  = 31.4159265 cm  (approximately).


Now the angle of the sector    is connected with the slant height and the arc length
by equation

    C = ,  

where    is in radians.  Here "h"  plays the role of the radius of the arc.


So, we find

     =  =  = 2.416609731 radians.


If you want to have it in degrees, convert radians to degrees

     =  =  = 138.4615 degrees (approximately).


ANSWER.  The angle of the sector is about 2.416609731 radians or 138.4615 degrees.