The length of each side of a regular hexagon is 12 cm. The length of one of the diagonals of a rhombus whose area is equal to $\sqrt{3}$ times the area of the hexagon is 72 cm. What is the length (in cm) of the other diagonal of the rhombus?

Area of a regular hexagon with side 12 cm is $\dfrac{3\sqrt{3}}{2}(12)^2=216\sqrt{3}$ square cm.

The area of the rhombus is $\sqrt{3}$ times the area of the hexagon:

$216\sqrt{3}\times\sqrt{3}=648$

If the rhombus diagonals are 72 cm and $d$ cm:

$\dfrac{1}{2}\times72\times d=648$

$36d=648$

$d=18$ cm.