The distance from a specified center point to points on a circle, sphere, or any radial figure; a nonnegative scalar that sets the scale of a centred metric object.

Principle

Radius encodes radial symmetry and scale: specifying a center and radius determines the set of points at that fixed distance, and many geometric formulas scale predictably with r.

Demonstration

A circle of radius r in the plane is the set of points at Euclidean distance r from its center; in Euclidean space the sphere of radius R has surface area proportional to R^2 and volume proportional to R^3.

Misapplication

Calling an arbitrary closest-point distance a 'radius' without a well-defined central point, or treating radius as directional (it is a scalar distance) in asymmetric or off-center contexts.

Consequence

Radius fixes circumference, area, and volume formulas for symmetric objects and provides a natural parameter for scaling, dilation, and comparison of sizes.

Reversal

Replacing center-to-boundary distance by maximal internal distance produces diameter; choosing the minimal radius of an enclosing ball yields circumradius or covering radius instead of an arbitrary radius.

Boundary

Requires a chosen center; for general sets there may be many radii relative to different centers. In non-metric or discrete contexts 'radius' must be defined via the relevant distance function.

Semantic Tension

Tension arises between informal uses (any characteristic length called 'radius') and strict metric usage (distance from a specific center); related notions include inradius, circumradius, and effective radius in nonuniform distributions.

Synthesis

Radius is the center-to-boundary distance that sets the scale of circular or spherical objects; as a metric quantity it organizes formulas, symmetries, and scaling behavior when a center is specified.