What is the length of the sides of a hexagon containing 19 x Ø2" billiard balls?

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I'm looking to make a hexagon wooden rack to hold 19 x Ø2" billiard balls. What is the length of the inside sides of the hexagon containing 19 x Ø2" billiard balls?

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As $AB=OB\tan(30^\circ)=1''\cdot\dfrac{\sqrt{3}}{3}$ and $BD=OO_1=2''$ then the side length will be $\dfrac{2(\sqrt{3}+6)}{3}\approx 5.1574 ('')$.