Inscribed Hexagon

Radius of inscribed circle and a side of the inscribed hexagon are the same, both equal $6\times cos(30^\circ)=6\times\frac{\sqrt{3}}{2}$ .

Areas are proportional to squares of distances, so we need the ratio of squares, which is $\frac{6^2}{6^2\times\frac{3}{4}}=\frac{4}{3}$ .

The radius of the inscribed circle is the side length of the smaller regular hexagon. Let $x$ be the radius and by pythagorean theorem,

$x = \sqrt{6^2 - 3^2} = \sqrt{27}$

ratio = $\frac{6^2}{(\sqrt{27})^2} = 1.33$