Circumscribed circle

Circumference of a polygon (red)

In geometry, the circumscribed circle is the circle that passes through all the vertices of a polygon and completely contains said figure in its interior. The center of the circumscribed circle is called circumcenter and its radius circumradius.

A polygon that has a circumcircle is called a cyclic polygon. All simple regular polygons, all triangles, and all rectangles are cyclic. In every cyclic polygon, the circumcenter is at the point of intersection of the perpendicular bisectors of the sides of the polygon.

Circumcircle of triangles

Triangles are the only polygons that are guaranteed to have a circumcircle, since they are always cyclic polygons. The only exception are some degenerate triangles that have a null surface.

Circle of quadrilaterals

Cyclical quadrilateral

Inscribed quadrilaterals have particular properties, including that opposite angles are supplementary which follows from the generalization of the capable arc.