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