In classical geometry, a radius (PL: radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latinradius, meaning ray but also the spoke of a chariot wheel.[1] The plural of radius can be either radii (from the Latin plural) or the conventional English plural radiuses.[2] The typical abbreviation and mathematical variable name for radius is R or r. By extension, the diameter D is defined as twice the radius:[3]
Circle with:
circumference C
diameter D
radius R
center or origin O
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.[4] 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.[5]
The radius of the circle that passes through the three non-collinear points P1, P2, and P3 is given by
where θ is the angle ∠P1P2P3. This formula uses the law of sines. If the three points are given by their coordinates (x1,y1), (x2,y2), and (x3,y3), the radius can be expressed as
The radius r of a regular polygon with n sides of length s is given by r = Rns, where Values of Rn for small values of n are given in the table. If s = 1 then these values are also the radii of the corresponding regular polygons.
Hypercubes
The radius of a d-dimensional hypercube with side s is
The fixed point (analogous to the origin of a Cartesian system) is called the pole, and the 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.[6]
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 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,[7] or axial position.[8]
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.
^Definition of Radius at dictionary.reference.com. Accessed on 2009-08-08.
^"Radius - Definition and More from the Free Merriam-Webster Dictionary". Merriam-webster.com. Retrieved 2012-05-22.
^ Definition of radius at mathwords.com. Accessed on 2009-08-08.
^Barnett Rich, Christopher Thomas (2008), Schaum's Outline of Geometry, 4th edition, 326 pages. McGraw-Hill Professional. ISBN0-07-154412-7, ISBN978-0-07-154412-2. Online version accessed on 2009-08-08.
^Jonathan L. Gross, Jay Yellen (2006), Graph theory and its applications. 2nd edition, 779 pages; CRC Press. ISBN1-58488-505-X, 9781584885054. Online version accessed on 2009-08-08.
^Brown, Richard G. (1997). Andrew M. Gleason (ed.). Advanced Mathematics: Precalculus with Discrete Mathematics and Data Analysis. Evanston, Illinois: McDougal Littell. ISBN0-395-77114-5.
^Krafft, C.; Volokitin, A. S. (1 January 2002). "Resonant electron beam interaction with several lower hybrid waves". Physics of Plasmas. 9 (6): 2786–2797. Bibcode:2002PhPl....9.2786K. doi:10.1063/1.1465420. ISSN 1089-7674. Archived from the original on 14 April 2013. Retrieved 9 February 2013. ...in cylindrical coordinates (r,θ,z) ... and Z=vbzt is the longitudinal position...
^Groisman, Alexander; Steinberg, Victor (1997-02-24). "Solitary Vortex Pairs in Viscoelastic Couette Flow". Physical Review Letters. American Physical Society (APS). 78 (8): 1460–1463. arXiv:patt-sol/9610008. Bibcode:1997PhRvL..78.1460G. doi:10.1103/physrevlett.78.1460. ISSN 0031-9007. S2CID 54814721. "[...]where r, θ, and z are cylindrical coordinates [...] as a function of axial position[...]"
January 10, 2023
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This article is about the line segment For the bone see Radius bone For other uses see Radius disambiguation In classical geometry a radius PL radii of a circle or sphere is any of the line segments from its center to its perimeter and in more modern usage it is also their length The name comes from the latin radius meaning ray but also the spoke of a chariot wheel 1 The plural of radius can be either radii from the Latin plural or the conventional English plural radiuses 2 The typical abbreviation and mathematical variable name for radius is R or r By extension the diameter D is defined as twice the radius 3 Circle with circumference C diameter D radius R center or origin O 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 4 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 5 The radius of the circle with perimeter circumference C is r C 2 p displaystyle r frac C 2 pi Contents 1 Formula 1 1 Circles 1 2 Regular polygons 1 3 Hypercubes 2 Use in coordinate systems 2 1 Polar coordinates 2 2 Cylindrical coordinates 2 3 Spherical coordinates 3 See also 4 ReferencesFormula EditFor many geometric figures the radius has a well defined relationship with other measures of the figure Circles Edit See also Area of a circle The radius of a circle with area A is r A p displaystyle r sqrt frac A pi The radius of the circle that passes through the three non collinear points P1 P2 and P3 is given by r O P 1 O P 3 2 sin 8 displaystyle r frac vec OP 1 vec OP 3 2 sin theta where 8 is the angle P1P2P3 This formula uses the law of sines If the three points are given by their coordinates x1 y1 x2 y2 and x3 y3 the radius can be expressed as r x 2 x 1 2 y 2 y 1 2 x 2 x 3 2 y 2 y 3 2 x 3 x 1 2 y 3 y 1 2 2 x 1 y 2 x 2 y 3 x 3 y 1 x 1 y 3 x 2 y 1 x 3 y 2 displaystyle r frac sqrt x 2 x 1 2 y 2 y 1 2 x 2 x 3 2 y 2 y 3 2 x 3 x 1 2 y 3 y 1 2 2 x 1 y 2 x 2 y 3 x 3 y 1 x 1 y 3 x 2 y 1 x 3 y 2 Regular polygons Edit See also Circumscribed circle n Rn3 0 577350 4 0 707106 5 0 850650 6 1 07 1 152382 8 1 306562 9 1 461902 10 1 618033 A square for example n 4 The radius r of a regular polygon with n sides of length s is given by r Rn s where R n 1 2 sin p n displaystyle R n 1 left left 2 sin frac pi n right right Values of Rn for small values of n are given in the table If s 1 then these values are also the radii of the corresponding regular polygons Hypercubes Edit The radius of a d dimensional hypercube with side s is r s 2 d displaystyle r frac s 2 sqrt d Use in coordinate systems EditPolar coordinates Edit Main article Polar coordinate system The polar coordinate system is a two dimensional coordinate system in which each point on a 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 system is called the pole and the 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 6 Cylindrical coordinates Edit 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 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 7 or axial position 8 Spherical coordinates Edit 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 EditBend radius Filling radius in Riemannian geometry Radius of convergence Radius of convexity Radius of curvature Radius of gyration SemidiameterReferences Edit Definition of Radius at dictionary reference com Accessed on 2009 08 08 Radius Definition and More from the Free Merriam Webster Dictionary Merriam webster com Retrieved 2012 05 22 Definition of radius at mathwords com Accessed on 2009 08 08 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 Online version accessed on 2009 08 08 Jonathan L Gross Jay Yellen 2006 Graph theory and its applications 2nd edition 779 pages CRC Press ISBN 1 58488 505 X 9781584885054 Online version accessed on 2009 08 08 Brown Richard G 1997 Andrew M Gleason ed Advanced Mathematics Precalculus with Discrete Mathematics and Data Analysis Evanston Illinois McDougal Littell ISBN 0 395 77114 5 Krafft C Volokitin A S 1 January 2002 Resonant electron beam interaction with several lower hybrid waves Physics of Plasmas 9 6 2786 2797 Bibcode 2002PhPl 9 2786K doi 10 1063 1 1465420 ISSN 1089 7674 Archived from the original on 14 April 2013 Retrieved 9 February 2013 in cylindrical coordinates r 8 z and Z vbzt is the longitudinal position Groisman Alexander Steinberg Victor 1997 02 24 Solitary Vortex Pairs in Viscoelastic Couette Flow Physical Review Letters American Physical Society APS 78 8 1460 1463 arXiv patt sol 9610008 Bibcode 1997PhRvL 78 1460G doi 10 1103 physrevlett 78 1460 ISSN 0031 9007 S2CID 54814721 where r 8 and z are cylindrical coordinates as a function of axial position Retrieved from https en wikipedia org w index php title Radius amp oldid 1127296008, wikipedia, wiki, book, books, library,
