Consider the angular bisectors of the inscribed triangle. Those bisectors meet at the center of the triangle. Any two of them along with a side of the triangle form a smaller isosceles triangle with base angles = 30 degrees and sides equal to the radius of the unit circle, which is 1 of course.
The remaining angle of that isosceles triangle is 180 - 2*30 = 120. We can then calculate the base of the isosceles triangle using the law of sines.
sin 30/1 = sin 120 /x => x = sin 120/sin 30 = sqrt 3;
Since this base is one side of an equilateral triangle, the perimeter of the triangle is then 3 * sqrt 3.