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

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

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

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

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

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

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

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

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.