##  [Radius](/radius-0) 

 Definition

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

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

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

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

Consequence

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

 

 

 

 

## Reversal

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

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

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

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.