{"id":455,"date":"2019-06-28T16:11:45","date_gmt":"2019-06-28T16:11:45","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=455"},"modified":"2019-12-20T22:59:04","modified_gmt":"2019-12-20T22:59:04","slug":"rolling-polygons","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/rolling-polygons\/","title":{"rendered":"Rolling Polygons"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

Perhaps you’ve learned from a\u00a0calculus\u00a0class 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. <\/p>\n\n\n\n

But did you know that a very similar fact is true for polygons?<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

Connect the red dots with straight lines<\/em>. (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.<\/p>\n\n\n\n

The same holds for pentagons, hexagons, and any regular n-gon!<\/p>\n\n\n\n

Presentation Suggestions:<\/strong>
Draw examples on the board! Challenge students to show this fact true for a triangle or a pentagon (harder).<\/p>\n\n\n\n

The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong>
Regular n-gons with a large number of sides are approximately circular, and the polygonal path obtained by connecting the dots will approximately\u00a0converge\u00a0to the path taken by a point on the edge of the disk! This recovers the result for the cycloid.<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Rolling Polygons.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

References:<\/strong>
pointed out to me by Duane deTemple<\/p>\n\n\n\n

Fun Fact suggested by: <\/strong>
Francis Su <\/p>\n","protected":false},"excerpt":{"rendered":"

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