Cyclogon

{{short description|Curve traced by a vertex of a polygon as it rolls}}

In geometry, a cyclogon is the curve traced by a vertex of a regular polygon that rolls without slipping along a straight line.{{cite book|last1=Tom M. Apostol, Mamikon Mnatsakanian|title=New Horizons in Geometry|url=https://archive.org/details/newhorizonsgeome00apos|url-access=limited|date=2012|publisher=Mathematical Association of America|isbn=9780883853542|page=[https://archive.org/details/newhorizonsgeome00apos/page/n82 68]}}{{cite web|last1=Ken Caviness|title=Cyclogons|url=http://demonstrations.wolfram.com/Cyclogons/|website=Wolfram Demonstrations Project|accessdate=23 December 2015}}

In the limit, as the number of sides increases to infinity, the cyclogon becomes a cycloid.{{cite journal|last1=T. M. Apostol and M. A. Mnatsakanian|title=Cycloidal Areas without Calculus|journal=Math Horizons|date=1999|volume=7|issue=1|pages=12–16|doi=10.1080/10724117.1999.12088451 |url=http://www.mamikon.com/USArticles/CycloidAreas.pdf|archive-url=https://web.archive.org/web/20050130091144/http://www.mamikon.com/USArticles/CycloidAreas.pdf|url-status=dead|archive-date=2005-01-30|accessdate=23 December 2015}}

The cyclogon has an interesting property regarding its area. Let {{mvar|A}} denote the area of the region above the line and below one of the arches, let {{mvar|P}} denote the area of the rolling polygon, and let {{mvar|C}} denote the area of the disk that circumscribes the

polygon. For every cyclogon generated by a regular polygon,

: $A\; =\; P\; +\; 2C.\; \backslash ,$