View source for Diameter

{{Short description|Straight line segment that passes through the centre of a circle}}
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[[File:Circle-withsegments.svg|thumb|right|Circle with {{legend-line|black solid 3px|[[circumference]] ''C''}}
{{legend-line|blue solid 2px|diameter ''D''}}
{{legend-line|red solid 2px|radius ''R''}}
{{legend-line|green solid 2px|centre or origin ''O''}}]]
{{General geometry}}
In [[Geometry|geometry]], a '''diameter''' of a [[Circle|circle]] is any straight [[Line segment|line segment]] that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest [[Chord (geometry)|chord]] of the circle. Both definitions are also valid for the diameter of a [[Sphere|sphere]].

In more modern usage, the length <math>d</math> of a diameter is also called the diameter. In this sense one speaks of {{em|the}} diameter rather than {{em|a}} diameter (which refers to the line segment itself), because all diameters of a circle or sphere have the same length, this being twice the [[Radius|radius]] <math>r.</math>

:<math>d = 2r \qquad\text{or equivalently}\qquad r = \frac{d}{2}.</math>

For a [[Convex set|convex]] shape in the [[Plane (geometry)|plane]], the diameter is defined to be the largest distance that can be formed between two opposite parallel lines [[Tangent|tangent]] to its boundary, and the {{em|width}} is often defined to be the smallest such distance. Both quantities can be calculated efficiently using [[Rotating calipers|rotating calipers]].<ref>{{cite web|author=Toussaint, Godfried T.|title=Solving geometric problems with the rotating calipers |publisher=Proc. MELECON '83, Athens|year=1983|citeseerx=10.1.1.155.5671}}</ref> For a [[Curve of constant width|curve of constant width]] such as the [[Reuleaux triangle]], the width and diameter are the same because all such pairs of parallel tangent lines have the same distance.

{{Anchor|Ellipse}}

For an [[Ellipse|ellipse]], the standard terminology is different. A diameter of an ellipse is any [[Chord (geometry)|chord]] passing through the centre of the ellipse.<ref>{{cite web|url=http://www.cut-the-knot.org/Curriculum/Geometry/ConjugateDiameters.shtml|title=Conjugate Diameters in Ellipse|first=Alexander|last=Bogomolny|website=www.cut-the-knot.org}}</ref> For example, [[Conjugate diameters|conjugate diameters]] have the property that a tangent line to the ellipse at the endpoint of one diameter is parallel to the conjugate diameter. The longest diameter is called the major axis.

The word "diameter" is derived from {{lang-grc|διάμετρος}} ({{transl|grc|diametros}}), "diameter of a circle", from {{lang|grc|διά}} ({{transl|grc|dia}}), "across, through" and {{lang|grc|μέτρον}} ({{transl|grc|metron}}), "measure".<ref>{{cite web|url=http://www.etymonline.com/index.php?term=diameter|title=diameter - Origin and meaning of diameter by Online Etymology Dictionary|website=www.etymonline.com}}</ref> It is often abbreviated <math>\text{DIA}, \text{dia}, d,</math> or <math>\varnothing.</math>

== Generalizations ==

The definitions given above are only valid for circles, spheres and convex shapes. However, they are special cases of a more general definition that is valid for any kind of <math>n</math>-dimensional (convex or non-convex) object, such as a [[Hypercube|hypercube]] or a [[Set (mathematics)|set]] of scattered points. The {{em|{{visible anchor|diameter|Diameter of a set|Diameter of a subset}}}} or {{em|{{visible anchor|metric diameter|Metric diameter of a set|Metric diameter of a subset}}}} of a [[Subset|subset]] of a [[Metric space|metric space]] is the least upper bound of the set of all distances between pairs of points in the subset. Explicitly, if <math>S</math> is the subset and if <math>\rho</math> is the [[Metric (mathematics)|metric]], the diameter is
<math display="block">\operatorname{diam}(S) = \sup_{x, y \in S} \rho(x, y).</math>

If the metric <math>\rho</math> is viewed here as having [[Codomain|codomain]] <math>\R</math> (the set of all [[Real number|real number]]s), this implies that the diameter of the [[Empty set|empty set]] (the case <math>S = \varnothing</math>) equals <math>- \infty</math> (negative infinity). Some authors prefer to treat the empty set as a special case, assigning it a diameter of <math>0,</math><ref>{{cite web|url=http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2004;task=show_msg;msg=0860.0002|title=Re: diameter of an empty set|website=at.yorku.ca}}</ref> which corresponds to taking the codomain of <math>d</math> to be the set of nonnegative reals.

For any solid object or set of scattered points in <math>n</math>-dimensional [[Euclidean space]], the diameter of the object or set is the same as the diameter of its [[Convex hull|convex hull]]. In medical parlance concerning a lesion or in geology concerning a rock, the diameter of an object is the least upper bound of the set of all distances between pairs of points in the object.

In [[Differential geometry|differential geometry]], the diameter is an important global [[Riemannian geometry|Riemannian]] [[Invariant (mathematics)|invariant]].

In planar geometry, a diameter of a [[Conic section|conic section]] is typically defined as any chord which passes through the conic's centre; such diameters are not necessarily of uniform length, except in the case of the circle, which has [[Eccentricity (mathematics)|eccentricity]] <math>e = 0.</math>

== Symbol ==
[[Image:Technical Drawing Hole 01.svg|thumb|122px|Sign ⌀ in a technical drawing]]
[[Image:Sign diameter.png|thumb|150px|Sign {{unichar|2205|EMPTY SET|ulink=Mathematical Operators}} from an AutoCAD drawing in dim.shx font with an angle 16°. This font does not contain {{unichar|2300|DIAMETER SIGN|ulink=Miscellaneous Technical}}.]]

The [[Symbol|symbol]] or [[Variable (mathematics)|variable]] for diameter, {{char|⌀}}, is sometimes used in technical drawings or specifications as a prefix or suffix for a number (e.g. "⌀ 55 mm"), indicating that it represents diameter. For example, photographic filter thread sizes are often denoted in this way.

In German, the diameter symbol (German ''Durchmesserzeichen'') is also used as an [[Average|average]] symbol (''Durchschnittszeichen'').

=== Encodings ===

The symbol has a [[Unicode]] [[Code point|code point]] at {{unichar|2300|Diameter sign|html=}}, in the [[Miscellaneous Technical]] set. On an [[Company:Apple Inc.|Apple]] [[Engineering:Macintosh|Macintosh]], the diameter symbol can be entered via the character palette (this is opened by pressing {{key press|Opt|Cmd|T|chain=}} in most applications), where it can be found in the Technical Symbols category. In Unix/Linux/ChromeOS systems, it is generated using {{key press|Ctrl|Shift|U}}&nbsp;{{key press|2|3|0|0|space|chain=}}. It can be obtained in [[Unix-like]] operating systems using a Compose key by pressing, in sequence, {{key press|Compose|d|i|chain=}}.<ref>{{cite web |url=http://cgit.freedesktop.org/xorg/lib/libX11/plain/nls/en_US.UTF-8/Compose.pre |title=UTF-8 (Unicode) compose sequence |last=Monniaux |first=David |access-date=2018-07-13}}</ref> In Windows, it can be entered in most programs with [[Alt code]] 8960.

The character will sometimes not display correctly, however, since many [[Typeface|font]]s do not include it. In many situations, the Nordic letter ø at Unicode {{unichar|00F8|Latin small letter o with stroke|html=}} is an acceptable substitute. It can be entered on a Macintosh by pressing {{key press|Opt|O|chain=}} (the letter o, not the number [[0|0]]). In Unix/Linux/ChromeOS systems, it is generated using {{key press|Ctrl|Shift|U}}&nbsp;{{key press|F|8|space|chain=}} or {{key press|Compose|o|/|chain=}}. [[Software:AutoCAD|AutoCAD]] uses {{unichar|2205|EMPTY SET|ulink=Mathematical Operators}} available as a shortcut string {{Kbd|%%c}}.

In [[Software:Microsoft Word|Microsoft Word]], the diameter symbol can be acquired by typing {{key press|2|3|0|0|chain=}} and then pressing {{key press|Alt|X}}.

In [[Software:LaTeX|LaTeX]], the diameter symbol can be obtained with the command <code>\diameter</code> from the "wasysym" package.<ref name="wasysym">{{cite web |url=https://ctan.org/pkg/wasysym |title=wasysym – LaTeX support for the wasy fonts |website=[[CTAN|Comprehensive TeX Archive Network]] |access-date=2022-03-11}}</ref>

== Diameter vs. radius ==
The diameter of a circle is exactly twice its radius. However, this is true only for a circle, and only in the [[Euclidean distance|Euclidean metric]]. [[Jung's theorem]] provides more general inequalities relating the diameter to the radius.

== See also ==
{{Div col|colwidth=30em}}
* {{annotated link|Astronomy:Angular diameter}}
* Caliper, micrometer, tools for measuring diameters
* {{annotated link|Conjugate diameters}}
* {{annotated link|Diameter (group theory)}}, a concept in [[Group theory|group theory]]
* [[Biography:Eratosthenes|Eratosthenes]], who calculated the diameter of the [[Astronomy:Earth|Earth]] around 240 BC.
* {{annotated link|Distance (graph theory)|Graph or network diameter}}
* {{annotated link|Hydraulic diameter}}
* {{annotated link|Sauter mean diameter}}
* {{annotated link|Tangent lines to circles}}
* The [[Engineering:Screw thread#Diameters|diameters]] of a screwthread
{{div col end}}

== References ==

{{reflist|group=note}}
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[[Category:Elementary geometry]]
[[Category:Length]]
[[Category:Circles]]

{{Sourceattribution|Diameter}}