View source for Radius

{{Short description|Segment in a circle or sphere from its center to its perimeter or surface and its length}}
{{About|the line segment|the bone|Radius (bone)||Radius (disambiguation)}}
[[File:Circle-withsegments.svg|thumb|right|Circle with:
{{legend-line|red solid 2px|radius ''R''}}
{{legend-line|black solid 3px|circumference ''C''}}
{{legend-line|blue solid 2px|diameter ''D''}}
{{legend-line|green solid 2px|center or origin ''O''}}]]

In classical [[geometry]], a '''radius''' ({{plural form}}: '''radii''' or '''radiuses'''){{efn|The plural of radius can be either ''radii'' (from the Latin plural) or the conventional English plural ''radiuses''.<ref>{{cite web|url=http://www.merriam-webster.com/dictionary/radius |title=Radius - Definition and More from the Free Merriam-Webster Dictionary |publisher=Merriam-webster.com |access-date=2012-05-22}}</ref>}} of a [[circle]] or [[sphere]] is any of the [[line segment]]s from its [[Centre (geometry)|center]] to its [[perimeter]], and in more modern usage, it is also their length. The radius of a [[regular polygon]] is the line segment or distance from its center to any of its [[Vertex (geometry)|vertices]]. The name comes from the [[Latin]] ''radius'', meaning ray but also the spoke of a chariot wheel.<ref name="radic">[http://dictionary.reference.com/browse/Radius Definition of Radius] at dictionary.reference.com. Accessed on 2009-08-08.</ref> The typical abbreviation and [[mathematical symbol]] for radius is ''R'' or ''r''. By extension, the [[diameter]] ''D'' is defined as twice the radius:<ref name="mwd1">
[http://www.mathwords.com/r/radius_of_a_circle_or_sphere.htm Definition of radius] at mathwords.com. Accessed on 2009-08-08.</ref>

:<math>d \doteq 2r \quad \Rightarrow \quad r = \frac d 2.</math>

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 polygon]]s, the radius is the same as its circumradius.<ref name="schaum">Barnett Rich, Christopher Thomas (2008), ''Schaum's Outline of Geometry'', 4th edition, 326 pages. McGraw-Hill Professional. {{isbn|0-07-154412-7}}, {{isbn|978-0-07-154412-2}}. [https://books.google.com/books?id=ab8lZG2yubcC  Online version] accessed on 2009-08-08.</ref>  The inradius of a regular polygon is also called [[apothem]]. In [[graph theory]], the [[radius (graph theory)|radius of a graph]] is the minimum over all vertices ''u'' of the maximum distance from ''u'' to any other vertex of the graph.<ref name="yel">Jonathan L. Gross, Jay Yellen (2006), ''Graph theory and its applications''. 2nd edition, 779 pages; CRC Press. {{isbn|1-58488-505-X}}, 9781584885054. [https://books.google.com/books?id=unEloQ_sYmkC Online version] accessed on 2009-08-08.</ref>

The radius of the circle with [[perimeter]] ([[circumference]]) ''C'' is

:<math>r = \frac C {2\pi}.</math>

== Formula ==

For many geometric figures, the radius has a well-defined relationship with other measures of the figure.

===Circles===
{{See also|Area of a circle}}

The radius of a circle with [[area]] {{math|''A''}} is
: <math>r = \sqrt{\frac{A}{\pi}}.</math>

The radius of the circle that passes through the three non-[[collinear]] points {{math|''P''<sub>1</sub>}}, {{math|''P''<sub>2</sub>}}, and {{math|''P''<sub>3</sub>}} is given by

: <math>r=\frac{|\vec{OP_1}-\vec{OP_3}|}{2\sin\theta},</math>

where {{mvar|θ}} is the angle {{math|∠''P''<sub>1</sub>''P''<sub>2</sub>''P''<sub>3</sub>}}. This formula uses the [[law of sines]].  If the three points are given by their coordinates {{math|(''x''<sub>1</sub>,''y''<sub>1</sub>)}}, {{math|(''x''<sub>2</sub>,''y''<sub>2</sub>)}}, and {{math|(''x''<sub>3</sub>,''y''<sub>3</sub>)}}, the radius can be expressed as

: <math> r = \frac {\sqrt{\bigl((x_2 - x_1)^2 + (y_2 - y_1)^2\bigr) \bigl((x_2 - x_3)^2 + (y_2 - y_3)^2\bigr) \bigl((x_3 - x_1)^2 + (y_3 - y_1)^2 \bigr)} }{ 2\bigl| x_1  y_2 + x_2 y_3 + x_3 y_1 - x_1 y_3 - x_2 y_1 - x_3 y_2\bigr| }.</math>

=== Regular polygons ===
{{See also|Circumscribed circle}}

{| class="wikitable floatright"
! {{mvar|n}}
! {{math|''R''<sub>''n''</sub>}}
|-
| 3 || {{gaps|0.577|350...}}
|-
| 4 || {{gaps|0.707|106...}}
|-
| 5 || {{gaps|0.850|650...}}
|-
| 6 || {{center|1=1}}
|-
| 7 || {{gaps|1.152|382...}}
|-
| 8 || {{gaps|1.306|562...}}
|-
| 9 || {{gaps|1.461|902...}}
|-
| 10 || {{gaps|1.618|033...}}
|}
[[File:Regular polygon radius square.png|thumb|A square, for example (''n''=4)]]

The radius {{mvar|r}} of a regular polygon with {{mvar|n}} sides of length {{mvar|s}} is given by {{math|1=''r'' = ''R''<sub>''n''</sub> ''s''}}, where <math>R_n = 1\left/\left(2 \sin \frac\pi n \right)\right. .</math>  Values of {{math|''R''<sub>''n''</sub>}} for small values of {{mvar|n}} are given in the table. If {{math|1=''s'' = 1}} then these values are also the radii of the corresponding regular polygons.

<!-- To add: radius from area, inradius from outradius, outradius from inradius -->

=== Hypercubes ===

The radius of a ''d''-dimensional [[hypercube]] with side ''s'' is 
:<math> r = \frac{s}{2}\sqrt{d}.</math>

== Use in coordinate systems ==

===Polar coordinates===
{{main article|Polar coordinate system}}
The polar coordinate system is a [[Plane (mathematics)|two]]-[[dimension]]al [[coordinate system]] in which each [[point (mathematics)|point]] on a [[plane (mathematics)|plane]] is determined by a [[distance]] from a fixed point and an [[angle]] from a fixed direction.

The fixed point (analogous to the origin of a [[Cartesian coordinate system|Cartesian system]]) is called the ''pole'', and the [[ray (geometry)|ray]] from the pole in the fixed direction is the ''polar axis''. The distance from the pole is called the ''radial coordinate'' or ''radius'', and the angle is the ''angular coordinate'', ''polar angle'', or ''[[azimuth]]''.<ref name="brown">{{Cite book| last = Brown| first = Richard G.| editor = Andrew M. Gleason| year = 1997| title = Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis| publisher = McDougal Littell| location = Evanston, Illinois| isbn = 0-395-77114-5| url-access = registration| url = https://archive.org/details/advancedmathemat00rich_0}}</ref>

===Cylindrical coordinates===
{{main article|Cylindrical coordinate system}}
In the cylindrical coordinate system, there is a chosen reference axis and a chosen reference plane perpendicular to that axis. The ''origin'' of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis.

The axis is variously called the ''cylindrical'' or ''longitudinal'' axis, to differentiate it from
the ''polar axis'', which is the [[Line (mathematics)#Ray|ray]] that lies in the reference plane, 
starting at the origin and pointing in the reference direction.

The distance from the axis may be called the ''radial distance'' or ''radius'', 
while the angular coordinate is sometimes referred to as the ''angular position'' or as the ''azimuth''.
The radius and the azimuth are together called the ''polar coordinates'', as they correspond to a two-dimensional polar coordinate system in the plane through the point, parallel to the reference plane.
The third coordinate may be called the ''height'' or ''altitude'' (if the reference plane is considered horizontal), 
''longitudinal position'',<ref>{{cite journal |last1=Krafft |first1=C. |last2=Volokitin |first2=A. S. |title=Resonant electron beam interaction with several lower hybrid waves |journal=Physics of Plasmas |date=1 January 2002 |volume=9 |issue=6 |pages=2786–2797 |doi=10.1063/1.1465420 |url=http://pop.aip.org/resource/1/phpaen/v9/i6/p2786_s1?isAuthorized=no |archive-url=https://archive.today/20130414005110/http://pop.aip.org/resource/1/phpaen/v9/i6/p2786_s1?isAuthorized=no |url-status=dead |archive-date=14 April 2013 |access-date=9 February 2013 |issn=1089-7674 |quote=...in cylindrical coordinates (''r'',''θ'',''z'') ... and Z=v<sub>bz</sub>t is the longitudinal position... |bibcode=2002PhPl....9.2786K }}</ref> 
or ''axial position''.<ref>{{cite journal | last1=Groisman | first1=Alexander | last2=Steinberg | first2=Victor | title=Solitary Vortex Pairs in Viscoelastic Couette Flow | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=78 | issue=8 | date=1997-02-24 | issn=0031-9007 | doi=10.1103/physrevlett.78.1460 | pages=1460–1463| arxiv=patt-sol/9610008 | bibcode=1997PhRvL..78.1460G | s2cid=54814721 }}  "<nowiki>[...]</nowiki>where ''r'', ''θ'', and ''z'' are cylindrical coordinates <nowiki>[...]</nowiki> as a function of axial position<nowiki>[...]</nowiki>"</ref>

===Spherical coordinates===
{{main article|Spherical coordinate system}}
In a spherical coordinate system, the radius describes the distance of a point from a fixed origin. Its position if further defined by the polar angle measured between the radial direction and a fixed zenith direction, and the azimuth angle, the angle between the orthogonal projection of the radial direction on a reference plane that passes through the origin and is orthogonal to the zenith, and a fixed reference direction in that plane.

==See also==
{{colbegin}}
*[[Bend radius]]
*[[Filling radius]] in Riemannian geometry
*[[Mean radius]]
*[[Radius of convergence]]
*[[Radius of convexity]]
*[[Radius of curvature]]
*[[Radius of gyration]]
*[[Semidiameter]]
{{Div col end}}

==Notes==
{{Notelist}}

==References==
{{Reflist|2}}

{{Authority control}}

[[Category:Radii|*]]