Diagonal Lengths

We note that we can cut off an end triangle of the trapezoid, flip it and place it on the other end, converting the trapezoid into a rectangle with the same area. Let the length of a diagonal be $d$ . Then the area of the rectangle is given by:

$\begin{aligned} \text{Base }\times \text{ Height } & = 36 \sqrt 3 \\ d \cos 30^\circ \times d \sin 30^\circ & = 36 \sqrt 3 \\ \frac {\sqrt 3}4 d^2 & = 36\sqrt 3 \\ d^2 & = 144 \\ \implies d & = \boxed{12} \end{aligned}$