Take a look at the interactivity below which shows regular polygons "rolling" along the horizontal surface.
It leaves a trace of the path of the red dot and on the graph it records the distance that the dot travels.
Experiment by positioning the red dot at the centre of the polygons, at one of the vertices or at the centre of one of the sides of the polygons and explore how this affects the distance / time graph.
Full Screen Version
Can you now work out what produced the following distance / time graph?
The rolling polygon had a radius of 40.
Can you work out how many sides it had and where the red dot was placed?
Try to explain how you worked it out.