radius

In classical geometry, the radius of a circle or sphere is the length of a line segment from its center to its perimeter. The name comes from Latin radius, meaning "ray" but also the spoke of a chariot wheel. The plural of radius can be either radii (from the Latin plural) or the conventional English plural radiuses. The typical abbreviation and mathematic variable name for "radius" is r. By extension, the diameter d is defined as twice the radius:




d
≐
2
r

⇒

r
=


d
2


.


{\displaystyle d\doteq 2r\quad \Rightarrow \quad r={\frac {d}{2}}.}

If an object does not have a center, the term may refer to its circumradius, the radius of its circumscribed circle or circumscribed sphere. In either case, the radius may be more than half the diameter, which is usually defined as the maximum distance between any two points of the figure. The inradius of a geometric figure is usually the radius of the largest circle or sphere contained in it. The inner radius of a ring, tube or other hollow object is the radius of its cavity.
For regular polygons, the radius is the same as its circumradius. The inradius of a regular polygon is also called apothem. In graph theory, the radius of a graph is the minimum over all vertices u of the maximum distance from u to any other vertex of the graph.
The radius of the circle with perimeter (circumference) C is




r
=


C

2
π



.


{\displaystyle r={\frac {C}{2\pi }}.}

Alternatively, this can be expressed as




r
=


C
τ


.


{\displaystyle r={\frac {C}{\tau }}.}

, with




τ



{\displaystyle {\tau }}
(tau) being equal to



2

π



{\displaystyle 2{\pi }}
exactly, although this has yet to gain mainstream usage.



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