Octagon unrolling along a line

There is a regular octagon unrolling along a horizontal line. You can recognize, that the area under the generated cycloid is three times the area of the octagon.
(1) Explain the idea of the proof by using the checkboxes! (2) Realize this fact by cutting papermodels! (3) Are you able to generalize the problem for n-gons? So you can fit this file by creating a n-slider.