In the figure to the right, a circle is circumscribed around a regular hexagon, and the same circle is inscribed within another regular hexagon.
Let be the perimeter of the larger hexagon, let be the perimeter of the smaller hexagon, and let be the circumference of the circle.
The circumference of the circle can be approximated by finding the mean of the two perimeters:
If is approximated using the approximation for circumference above, then , where are positive integers, and are coprime, and is square-free.
Find .