Area and arc length of prolate cycloid loops

[liometopum]

Does anyone know of solutions (or sources showing them) to the area and line lengths of the loops and arcs of prolate cycloids? Standard cycloids are well discussed online, but equations relating to looping cycloid curves appear absent. I have not seen an equation relating to either loop area or arc area, for example.

 

[jedishrfu]

couldn't you compute this by noting that the circle doesn't slip so one complete cycle must trace out a circumference of the same length to match it for starters?

 

[liometopum]

The circle does slip in the case of prolate cycloids, creating the loops. And it 'skids' for curtate cycloids.