Let $x$ denote the common side length of the rhombi.
Let $y$ denote one of the smaller interior [angles](https://artofproblemsolving.com/wiki/index.php/Angle) of rhombus $\mathcal{P}$. Then $x^2\sin(y)=\sqrt{2006}$.  We also see that $K=x^2\sin(2y) \Longrightarrow K=2x^2\sin y \cdot \cos y \Longrightarrow K = 2\sqrt{2006}\cdot \cos y$.  Thus $K$ can be any positive integer in the [interval](https://artofproblemsolving.com/wiki/index.php/Interval) $(0, 2\sqrt{2006})$. 
$2\sqrt{2006} = \sqrt{8024}$ and $89^2 = 7921 < 8024 < 8100 = 90^2$, so $K$ can be any [integer](https://artofproblemsolving.com/wiki/index.php/Integer) between 1 and 89, inclusive.  Thus the number of positive values for $K$ is $\boxed{089}$.
[Hexagon](https://artofproblemsolving.com/wiki/index.php/Hexagon) $ABCDEF$ is divided into five [rhombuses](https://artofproblemsolving.com/wiki/index.php/Rhombus), $\mathcal{P, Q, R, S,}$ and $\mathcal{T,}$ as shown. Rhombuses $\mathcal{P, Q, R,}$ and $\mathcal{S}$ are [ congruent](https://artofproblemsolving.com/wiki/index.php/Congruent_(geometry)), and each has [area](https://artofproblemsolving.com/wiki/index.php/Area) $\sqrt{2006}.$ Let $K$ be the area of rhombus $\mathcal{T}$.  Given that $K$ is a [positive integer](https://artofproblemsolving.com/wiki/index.php/Positive_integer), find the number of possible values for $K$.