Regular Polygon

Pronunciation: /ˈrɛg yə lər ˈpɒl iˌgɒn/ Explain

A regular polygon is a polygon whose sides are equal length and whose sides are symmetrical about the center of the polygon.[1] Regular polygons of various number of sides can be denoted as 'regular n-gon'. A regular 3-gon is also called an equilateral triangle. A regular 4-gon is called a square.

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Manipulative 1 - Regular Polygons Created with GeoGebra.

Center of Regular Polygons

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Why can you not use the perpendicular bisectors of sides that are opposite each other to find the center of a regular polygon? Hint: Try it on a hexagon.
Manipulative 2 - Center of a Regular Polygon Created with GeoGebra.

Each regular polygon has a center. This center can be found by constructing the perpendicular bisectors of any two sides of the regular polygon. For polygons with an even number of sides, it can be found by connecting any two sets of antipodal (opposite) points.

The center of a regular polygon can be constructed by constructing the perpendicular bisector of two sides. The point of intersection of the perpendicular bisectors is the center of the regular polygon.

Central Angle of Regular Polygons

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Why, when there are more sides, is the central angle smaller?
Manipulative 3 - Central Angle of a Regular Polygon Created with GeoGebra.

The central angle of a regular polygon is the angle between two rays that go from the center of the regular polygon and pass through two adjacent vertices of the polygon. The measure of the central angle of regular polygons is 360 degrees/n or 2 pi radians/n where n is the number of vertices of the polygon. Click on the blue points in manipulative 3 and drag them to change the figure.

Circumcircle About Regular Polygons

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Manipulative 4 - Circumcircle About a Regular Polygon Created with GeoGebra.

A circle can be drawn around every regular polygon that intercepts all the vertices of the polygon and none of the sides. This is the circumcircle of the regular polygon. The center of the regular polygon is also the circumcenter of the regular polygon. Click on the blue points in manipulative 4 and drag them to change the figure.

To construct the circumcircle about a regular polygon, place the point of the compass at the center of the polygon, and the stylus on a vertex, then draw the circle.

Incircle of Regular Polygons

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Manipulative 5 - Incircle of a Regular Polygon Created with GeoGebra.

A circle can be drawn inside every regular polygon that intercepts each of the sides of the polygon exactly once. This is the incircle of the regular polygon. The center of the regular polygon is also the incenter of the regular polygon. Click on the blue points in manipulative 5 and drag them to change the figure.

To construct the incircle of a regular polygon, construct the midpoint of any of the sides. Then place the point of the compass at the center of the polygon, and the stylus on the midpoint, then draw the circle.

References

  1. Stöcker, K.H.. The Elements of Constructive Geometry, Inductively Presented. pg 28. Translated by Noetling, William A.M, C.E.. www.archive.org. Silver, Burdett & Company. 1897. Last Accessed 12/4/2018. http://www.archive.org/stream/elementsofconstr00noetrich#page/28/mode/1up. Buy the book
  2. Convex. ams.org. Geometry Glossary. American Mathematical Society. Last Accessed 12/4/2018. http://www.ams.org/featurecolumn/archive/geometry-glossary.html.

More Information

  • McAdams, David E.. Polygon. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 3/12/2009. http://www.allmathwords.org/en/p/polygon.html.

Cite this article as:

McAdams, David E. Regular Polygon. 12/5/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/r/regularpolygon.html.

Image Credits

Revision History

12/5/2019: Removed broken links, updated license, implemented new markup, implemented new Geogebra app. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
12/31/2009: Added "References". (McAdams, David E.)
12/31/2008: Changed equations from HTML to images. (McAdams, David E.)
12/11/2008: Added 'Center of a Regular Polygon'. Changed circumcircle figure to manipulative. Added 'Incircles of Regular Polygons' (McAdams, David E.)
11/2/2008: Changed manipulative to GeoGebra. (McAdams, David E.)
6/11/2008: Added section on the central angle of a regular polygon and circumcircle. (McAdams, David E.)
4/18/2008: Initial version. (McAdams, David E.)

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