Question 717989
Realize, a 30-60-90 triangle is half of another equilateral triangle.


Use the circle's circumference to get it radius.  The distance from the center of the circle to a midpoint of one of the sides of the triangle is a radius length and is one leg of a 30-60-90 triangle; the hypotenuse of this 30-60-90 triangle is TWICE the radius length.  

Now, one half the length of the circumscribed equilateral triangle is the other leg of the 30-60-90 triangle.  Use pythagorean theorem to find it and so if you multiply it by 2, you have the length of a side of the circumscribed equilateral triangle.  


Back a little, you get radius r; you then have 2r, and 2r is hypotenuse of a 30 60 90 triangle.  If y is hyptonuse for the 30 60 90 then {{{y^2=r^2+(2r)^2}}}.  Find y.
6y is the perimeter you want.