Circumcenter
Pronunciation: /ˈsɜr kəmˌsɛn tər/ ?
 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

 Pick any one side of a triangle and construct its perpendicular bisector.

2

 Pick one of the remaining sides of a triangle and construct its perpendicular bisector.

3

 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
Step  Illustration  Discussion and Justification 
1 

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 

Draw the perpendicular bisector of any side. 
3 

Draw the perpendicular bisector of any other side that is not opposite the side you used in step 2. 
4 

Label the intersection of the two perpendicular bisectors as 'center'. 
5 

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. 20090312. http://www.allmathwords.org/article.aspx?lang=en&id=Center.
 McAdams, David. Circle. allmathwords.org. Life is a Story Problem LLC. 20090312. http://www.allmathwords.org/article.aspx?lang=en&id=Circle.
 McAdams, David. Triangle. allmathwords.org. Life is a Story Problem LLC. 20090312. http://www.allmathwords.org/article.aspx?lang=en&id=Triangle.
Cite this article as:
Circumcenter. 20110428. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/circumcenter.html.
Translations
Image Credits
Revision History
20110428: Added circumradius. (
McAdams, David.)
20101013: 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.)
20100109: Added "References" (
McAdams, David.)
20081115: Changed manipulative to Geogebra (
McAdams, David.)
20080707: Corrected link errors. Corrected spelling (
McAdams, David.)
20080325: Revised More Information to match current standard (
McAdams, David.)
20070824: Simplified figure 1. Added reference to triangle. Added circumcircle (
McAdams, David.)
20070730: Initial version (
McAdams, David.)