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 ngon'.
A regular 3gon is also called an
equilateral triangle.
A regular 4gon is called a
square.
Click on the blue points and drag them to change the figure.

Manipulative 1  Regular Polygons Created with GeoGebra. 
Center of Regular Polygons
Click on the blue points and drag them to change the figure.
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
or
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
Click on the blue points and drag them to change the figure.
 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
 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
 Convex. ams.org. Geometry Glossary. American Mathematical Society. Last Accessed 12/4/2018. http://www.ams.org/featurecolumn/archive/geometryglossary.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.)