##  [Circumradius](/circumradius-0) 

 Definition

The radius of a circumscribed circle or sphere that passes through all vertices of a polygon or polyhedron; when a unique circumcenter exists, the circumradius is the distance from that center to any vertex.

 

 

 

 

 

 





## Principle

Principle

Circumradius depends on concurrency of perpendicular bisectors (circumcenter existence) and captures the scale of a cyclic configuration; in triangles it relates algebraically to side lengths and area.

 

 

 

 

 





## Demonstration

Demonstration

A triangle with side lengths a,b,c has circumradius R = abc/(4Δ) where Δ is the triangle's area; a regular n-gon of side s has a circumradius determined by s and the central angle.

 

 

 

 

## Misapplication

Misapplication

Assuming every polygon or polyhedron has a circumradius; many polygons are not cyclic and many polyhedra have no sphere through all vertices. Confusing circumradius with minimal enclosing ball radius.

 

 

 

 

 





## Consequence

Consequence

When it exists, circumradius fixes central angles, chord lengths, and is used in constructions, triangulations (Delaunay), and in deriving metric relations among vertices.

 

 

 

 

## Reversal

Reversal

The dual notion is inradius (inscribed radius) measured from an incenter to tangent sides; replacing 'passes through vertices' by 'is tangent to edges' swaps circumradius for inradius.

 

 

 

 

 





## Boundary

Boundary

Defined only for cyclic polygons and vertex-concyclic polyhedra; for arbitrary point sets one may instead consider the minimal enclosing radius (covering radius), which need not be realized by a single circumcenter equidistant to all points.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension between existence and optimality: circumradius requires exact concyclicity while minimal enclosing radius gives an optimal covering even when no true circumcircle exists; the two coincide only in special cases.

 

 

 

 

 





## Synthesis

Synthesis

Circumradius is the center-to-vertex distance of a circumscribed circle or sphere when such a circumscription exists; it quantifies the scale of a cyclic configuration and underpins many metric relations in polygonal and polyhedral geometry.