Approximating \(\pi\) with hexagons

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 \(P_1\) be the perimeter of the larger hexagon, let \(P_2\) be the perimeter of the smaller hexagon, and let \(C\) be the circumference of the circle.

The circumference of the circle can be approximated by finding the mean of the two perimeters: \[C\approx \frac{P_1+P_2}{2}.\] If \(\pi\) is approximated using the approximation for circumference above, then \(\displaystyle\pi\approx\frac{a}{b}+\sqrt{c}\), where \(a, b, c\) are positive integers, \(a\) and \(b\) are coprime, and \(c\) is square-free.

Find \(a+b+c\).