Approximating $$\pi$$ with hexagons

Geometry Level 4

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 $$a+b+c$$.

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