2013 AMC 10B Problem 15

Let $s$ be the side length of the equilateral triangle and $A$ its area. Then $a=3s.$ A regular hexagon of side length $s$ consists of $6$ equilateral triangles of side length $s,$ so its area is $6A.$ Therefore, a regular hexagon of area $A$ has side length $\dfrac{s}{\sqrt 6} .$ Hence $$b = 6\cdot \dfrac{s}{\sqrt 6} = s\sqrt 6 .$$ It follows that $$\dfrac ab = \dfrac{3s}{s\sqrt 6} = \dfrac{3 \sqrt 6}6 = \dfrac{\sqrt 6} 2.$$ Thus, the correct answer is B.