Beautiful Geometry

In the above image we have five circles: The white one with radious of 4, a blue one, a green one, a red one and a pink one, all four congruent to each other. If the area enclosed by arcs AB (green), BC (pink), CD (blue) and DA (white) can be written like $\frac{a\sqrt{3}}{b} - \frac{cπ}{d}$ Find the minimum value of $a+b+c+d$ .

Assume $a,b,c,d$ are positive integers.