View source for Diameter

{{Short description|Straight line segment that passes through the centre of a circle}}
{{Other uses}}
{{use British English|date=March 2021}}<!-- so centre not center -->
[[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]], a '''diameter''' of a [[circle]] is any straight [[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]].

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]] <math>r.</math>

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

The word "diameter" is derived from {{langx|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 |title=Diameter—Origin and meaning of diameter by Online Etymology Dictionary |url=http://www.etymonline.com/index.php?term=diameter |website=www.etymonline.com}}</ref> It is often abbreviated <math>\text{DIA}, \text{dia}, d,</math> or <math>\varnothing.</math>

== Constructions ==
With [[straightedge and compass]], a diameter of a given circle can be constructed as the [[perpendicular bisector]] of an arbitrary chord. Drawing two diameters in this way can be used to locate the center of a circle, as their crossing point.<ref>{{cite book|title=General Drafting|series=Technical manual, TM 5-581A, United States Department of the Army|publisher=U.S. Government Printing Office|year=1972|page=6{{hyphen}}32|contribution=6-66 Finding the center of a circle|contribution-url=https://books.google.com/books?id=1iGqD3y_CggC&pg=SA6-PA32}}</ref> To construct a diameter parallel to a given line, choose the chord to be perpendicular to the line.

The circle having a given line segment as its diameter can be constructed by straightedge and compass, by finding the [[midpoint]] of the segment and then drawing the circle centered at the midpoint through one of the ends of the line segment.

== Symbol ==
{{redirect|⌀}}
[[Image:Technical Drawing Hole 01.svg|thumb|upright 0.5|Sign ⌀ in a technical drawing]]
[[File:Hama UV Filter 1.jpg|thumb|A [[photographic filter]] marked as having a 58mm thread diameter]]
The [[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.<ref>{{cite book|title=Interpretation of Geometric Dimensioning and Tolerancing|first=Daniel E.|last=Puncochar|publisher=Industrial Press Inc.|year=1997|isbn=9780831130725|page=5|url=https://books.google.com/books?id=76la9qNmt24C&pg=PA5}}</ref> Photographic [[filter thread]] sizes are often denoted in this way.<ref>{{cite book|title=Introduction to Digital Photography|first=Joseph|last=Ciaglia|publisher=Prentice Hall|year=2002|isbn=9780130321367|page=9|quote=The filter diameter (in mm) usually follows the symbol ⌀}}</ref>

The symbol has a [[code point]] in [[Unicode]] at {{unichar|2300|Diameter sign|html=}}, in the [[Miscellaneous Technical]] set. It should not be confused with several other characters (such as {{unichar|00D8|nlink=}} or {{unichar|2205|nlink=Null sign}}) that resemble it but have unrelated meanings.<ref>{{cite book|title=Unicode Explained|first=Jukka K.|last=Korpela|publisher=O'Reilly Media, Inc.|year=2006|isbn=9780596101213|page=171|url=https://books.google.com/books?id=lxndiWaFMvMC&pg=PA171}}</ref> It has the [[compose sequence]] {{key press|[[Compose key|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>

== Generalizations ==
The definitions given above are only valid for circles and spheres. However, they are special cases of a more general definition that is valid for any kind of <math>n</math>-dimensional object,  or a [[set (mathematics)|set]] of scattered points. The [[diameter of a set]] is the [[supremum|least upper bound]] of the set of all distances between pairs of points in the subset.

{{Anchor|Ellipse}}A different and incompatible definition is sometimes used for the diameter of a [[conic section]]. In this context, a diameter is any [[chord (geometry)|chord]] which passes through the [[center (geometry)#Projective conics|conic's centre]]. A diameter of an [[ellipse]] is any line 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> Half of any such diameter may be called a ''semidiameter'', although this term is most often a synonym for the [[radius]] of a circle or sphere.<ref>{{cite book|title=A Mathematical Dictionary|first1=Joseph|last1=Raphson|first2=Jacques|last2=Ozanam|publisher=J. Nicholson, and T. Leigh and D. Midwinter|year=1702|page=26|url=https://books.google.com/books?id=RMnn_islYKkC&pg=PA26}}</ref> The longest diameter is called the [[major axis]]. [[Conjugate diameters]] are a pair of diameters where one is parallel to a tangent to the ellipse at the endpoint of the other diameter.

Several kinds of object can be measured by [[equivalent diameter]], the diameter of a circular or spherical approximation to the object. This includes [[hydraulic diameter]], the equivalent diameter of a channel or pipe through which liquid flows, and the [[Sauter mean diameter]] of a collection of particles.

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}}
* [[Caliper]], [[Micrometer (device)|micrometer]], tools for measuring diameters
* [[Eratosthenes]], who calculated the diameter of the [[Earth]] around 240 BC.
* {{annotated link|Tangent lines to circles}}
{{div col end}}

== References ==

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{{Wiktionary|diameter}}
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[[Category:Elementary geometry]]
[[Category:Length]]
[[Category:Circles]]