Approximating π\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 P1P_1 be the perimeter of the larger hexagon, let P2P_2 be the perimeter of the smaller hexagon, and let CC be the circumference of the circle.

The circumference of the circle can be approximated by finding the mean of the two perimeters:

CP1+P22.C\approx \frac{P_1+P_2}{2}.

If π\pi is approximated using the approximation for circumference above, then πab+c\pi\approx\frac{a}{b}+\sqrt{c}, where a,b,ca, b, c are positive integers, aa and bb are coprime, and cc is square-free.

Find a+b+ca+b+c.


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