The circumcircle or circumscribed circle of a polygon is a circle which passes through all the vertices of the polygon. The center of this circle is called the circumcenter.

1. In triangles

In the case of a triangle, there is always a circumcircle possible, no matter what shape the triangle is. In the figure on the right, the red circle is the circumcircle of the triangle.

The center of the circumcircle is called the circumcenter, which may lie outside the triangle. See Circumcenter of a Triangle

For more on triangle circumcircles see Circumcircle of a Triangle.

It is possible to construct the circumcircle with a compass and straightedge. See Constructing the Circumcircle and Circumcenter of a Triangle

2. In regular polygons

Regular polygons, (polygons that have all sides the same length and all interior angles congruent) can have circumcircles. The center of the circumcircle, the circumcenter, is also considered to be the center of the polygon itself, since it is equidistant from each vertex.

For more on this see Circumcircle of a Regular Polygon and Regular Polygon definition.

Irregular Polygons

Irregular polygons are not thought of as having an circumcircle or even a center. If you were to draw a polygon at random, it is unlikely that there is a circle that passes through every vertex. An exception is the 3-sided polygon (triangle). All triangles always have a circumcircle (see above).

Pages referring to 'circumcircle'

Definition of the circumcircle of any triangle or regular polygon.
The circumcircle of a polygon is defined: the circle that passes through every vertex.
Circumcircle of a triangle. Definition and properties with interactive applet.
This page shows how to construct (draw) the circumcircle of a triangle with compass and straightedge or ruler. The circumcircle of a triangle is the circle that passes through all three vertices of the triangle. It's center is called the circumcenter, which is the point where the three perpedicular bisectors of the sides intersect. A Euclidean construction.
Given three points, it is always possible to draw a circle that passes through all three. This page shows how to construct (draw) a circle through 3 given points with compass and straightedge or ruler. It works by joining two pairs of points to create two chords. The perpendicular bisectors of a chords always passes through the center of the circle. By this method we find the center and can then draw the circle. A euclidean construction.
The radius of a regular polygon and formula for finding it.