Rectoid-ish
The height of parallel ceiling and floor is the diagonal of the non-degenerate rectangle. As shown, the chosen vertex draws its path as the rectangle rolls alternately along the ceiling and the floor. The region is formed with respect to the position of the chosen vertex at a start.
If either of the chosen right-sided vertices forms the region in the similar manner, what can be said about the area of the region in each of the four setups?
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1 solution
The path traced by the top (resp. bottom) left vertex becomes the path traced by the top (resp. bottom) right vertex by a reflection about a vertical line. Similarly, the path traced by the bottom left vertex becomes the path traced by the top left vertex by a reflection about a horizontal line followed by a horizontal translation along the sidelength of the square.
Since these paths define the regions and both reflections and translations are isometries, all four regions have the same area.