Circumcenter

Pronunciation: /ˈsɜr kəmˌsɛn tər/ ?

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
Manipulative 1: Circumcenter of a triangle. Created with GeoGebra.

The circumcenter of a polygon is the center of the circle that intersects all vertices of the polygon exactly once. The circumcircle of a polygon is the circle that intersects all vertices of the polygon. The incenter of a triangle is found at the intersection of the perpendicular bisectors of the sides. A circumradius of a polygon is a radius of the circumcircle.

How to Construct the Circumcenter and Circumcircle of a Triangle

1 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Pick any one side of a triangle and construct its perpendicular bisector.
2 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Pick one of the remaining sides of a triangle and construct its perpendicular bisector.
3 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Draw a circle with the center at the intersection of the two bisectors and a radius of the distance between the intersection and any vertex of the triangle.

How to Construct the Center and Circumcircle of a Regular Polygon

StepIllustrationDiscussion and Justification
1 A regular hexagon The center of a regular polygon is at the point of concurrency of perpendicular bisectors of any two sides that are not opposite each other.
2 A regular hexagon with the perpendicular bisector of one of the sides drawn in. Draw the perpendicular bisector of any side.
3 A regular hexagon with the perpendicular bisector of two of the non-opposite sides drawn in. Draw the perpendicular bisector of any other side that is not opposite the side you used in step 2.
4 A regular hexagon with the perpendicular bisector of two of the non-opposite sides drawn in. The intersection of the two perpendicular bisectors in labeled 'center'. Label the intersection of the two perpendicular bisectors as 'center'.
5 A regular hexagon with the perpendicular bisector of two of the non-opposite sides drawn in. The intersection of the two perpendicular bisectors in labeled 'center'. A circle is drawn with the center at 'center' and the edge at any vertex. Draw a circle with a center at the point labeled 'center' and the edge at any vertex. This is the circumcircle.
Table 2 - How to construct the center and circumcircle of a regular polygon

More Information

  • McAdams, David. Center. allmathwords.org. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Center.
  • McAdams, David. Circle. allmathwords.org. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Circle.
  • McAdams, David. Triangle. allmathwords.org. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Triangle.

Cite this article as:


Circumcenter. 2011-04-28. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/circumcenter.html.

Translations

Image Credits

Revision History


2011-04-28: Added circumradius. (McAdams, David.)
2010-10-13: Generalize article to deal with all polygons, rather than just triangles. Added section on constructing the circumcenter and circumcircle of a regular polygon. (McAdams, David.)
2010-01-09: Added "References" (McAdams, David.)
2008-11-15: Changed manipulative to Geogebra (McAdams, David.)
2008-07-07: Corrected link errors. Corrected spelling (McAdams, David.)
2008-03-25: Revised More Information to match current standard (McAdams, David.)
2007-08-24: Simplified figure 1. Added reference to triangle. Added circumcircle (McAdams, David.)
2007-07-30: Initial version (McAdams, David.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2005-2011 Life is a Story Problem LLC. All rights reserved.
Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License