Rolling Hexagon

The area enclosed by the path of one of the vertices of the rolling hexagon can be partitioned as follows:

The green sectors are each $\frac{1}{3}$ of a circle with a radius of $\sqrt{3}$ and therefore an area of $\frac{1}{3}\pi(\sqrt{3})^2 = \pi$ .

The purple sector is $\frac{1}{3}$ of a circle with a radius of $2$ and therefore an area of $\frac{1}{3}\pi(2)^2 = \frac{4}{3}\pi$ .

The red sectors are each $\frac{1}{3}$ of a circle with a radius of $1$ and therefore an area of $\frac{1}{3}\pi(1)^2 = \frac{1}{3}\pi$ .

All the sectors combine for an area of $\pi + \pi + \frac{4}{3}\pi + \frac{1}{3}\pi + \frac{1}{3}\pi = 4\pi$ , which is equivalent to the area of $4$ unit circles, so $X = 4$ .

The $4$ dark blue triangles can be rearranged into $1$ hexagon with unit sides, as shown below:

This hexagon, along with the light blue hexagon, makes $2$ hexagons total, so $Y = 2$ .

Since $X = 4$ and $Y = 2$ , $XY = \boxed{8}$ .