Rate of forward motion from rolling rectangle vs degrees turned

1. The problem statement, all variables and given/known data

I’m looking for a formula that relates the distance traveled by a ‘rolling’ rectangle compared to a given amount of degrees it has turned.

And while I have you, another problem I have with the rolling rectangle is that its center point would move up and then down with each 90 degree turn forward as it arches over its corner. What formula might describe this change in y-position of the rectangle’s center as it ‘rolls’ forward?

2. Relevant equations
See my attempt.

3. The attempt at a solution
The best I can come up with for the rolling rectangle’s forward motion is:
Δ (x position) = Δ (rotational position) [itex]\times[/itex] (b + (a-b)/2) where b is the new "bottom" side of the rectangle after a 90 degree roll, and a is the old "bottom" side. Although this seems like too linear of a formula to me. PI has got to work in there somewhere, right?

And for the Δy of the rectangle as it bobs along… the best I can discern from my sketches is Δy = Δrotational position [itex]\times[/itex] ((a-b)/2) for ascending segments and the inverse for the descending segments. Again, seems too linear.

Thanks for any insight. My first post here.


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