Perhaps you’ve learned from a calculus class that as you roll a circular disk along a straight line, that the area under the cycloid swept out by following a point on the edge of the disk between two successive points of tangency is exactly 3 times the area of the disk.
But did you know that a very similar fact is true for polygons?
For instance, take a square on a flat line, and mark one corner on the line with a red dot. Now “roll” it along the line by pivoting the square around the corner that touches the line. Each time it comes to a rest, mark the position of the red dot. When the red dot again touches the line, stop.
Connect the red dots with straight lines. (These are dotted lines in the Figure.) The area under this polygonal region will be 3 times the area of the square. You can verify this in Figure 1.
The same holds for pentagons, hexagons, and any regular n-gon!
Presentation Suggestions:
Draw examples on the board! Challenge students to show this fact true for a triangle or a pentagon (harder).
The Math Behind the Fact:
Regular n-gons with a large number of sides are approximately circular, and the polygonal path obtained by connecting the dots will approximately converge to the path taken by a point on the edge of the disk! This recovers the result for the cycloid.
How to Cite this Page:
Su, Francis E., et al. “Rolling Polygons.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.
References:
pointed out to me by Duane deTemple
Fun Fact suggested by:
Francis Su