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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
. In more modern usage, the length d of a diameter is also called the diameter. In this sense one speaks of diameter rather than 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 r. :d = 2r \qquad\text\qquad r = \frac. For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the is often defined to be the smallest such distance. Both quantities can be calculated efficiently using
rotating calipers In computational geometry, the method of rotating calipers is an algorithm design technique that can be used to solve optimization problems including finding the width or diameter of a set of points. The method is so named because the idea is ana ...
. For a
curve of constant width In geometry, a curve of constant width is a simple closed curve in the plane whose width (the distance between parallel supporting lines) is the same in all directions. The shape bounded by a curve of constant width is a body 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. For an ellipse, the standard terminology is different. A diameter of an ellipse is any chord passing through the centre of the ellipse. For example, 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 In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the ...
. The word "diameter" is derived from grc, διάμετρος (), "diameter of a circle", from (), "across, through" and (), "measure". It is often abbreviated \text, \text, d, or \varnothing.


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 n-dimensional (convex or non-convex) object, such as a hypercube or a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of scattered points. The or of a subset of a metric space is the least upper bound of the set of all distances between pairs of points in the subset. Explicitly, if S is the subset and if \rho is the metric, the diameter is \operatorname(S) = \sup_ \rho(x, y). If the metric \rho is viewed here as having codomain \R (the set of all real numbers), this implies that the diameter of the empty set (the case S = \varnothing) equals - \infty (
negative infinity In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra o ...
). Some authors prefer to treat the empty set as a special case, assigning it a diameter of 0, which corresponds to taking the codomain of d to be the set of nonnegative reals. For any solid object or set of scattered points in n-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, the diameter of the object or set is the same as the diameter of its 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, the diameter is an important global Riemannian invariant. In planar geometry, a diameter of a 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 e = 0.


Symbol

The symbol or variable for diameter, , 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 German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...
, the diameter symbol (German '' Durchmesserzeichen'') is also used as an average symbol (''Durchschnittszeichen'').


Similar symbols

The Latin small letter o with stroke is similar in size and design to this. The diameter symbol ⌀ is distinct from the empty set symbol , from an ( italic) uppercase phi , and from the Nordic vowel ( Latin capital letter O with stroke). See also slashed zero.


Encodings

The symbol has a Unicode code point at , in the Miscellaneous Technical set. On an Apple Macintosh, the diameter symbol can be entered via the character palette (this is opened by pressing in most applications), where it can be found in the Technical Symbols category. In Unix/Linux/ChromeOS systems, it is generated using  . It can be obtained in Unix-like operating systems using a Compose key by pressing, in sequence, . In Windows, it can be entered in most programs with Alt code 8960. The character will sometimes not display correctly, however, since many fonts do not include it. In many situations, the Nordic letter ø at Unicode is an acceptable substitute. It can be entered on a Macintosh by pressing (the letter o, not the number 0). In Unix/Linux/ChromeOS systems, it is generated using   or . AutoCAD uses available as a shortcut string . In Microsoft Word, the diameter symbol can be acquired by typing and then pressing . In LaTeX, the diameter symbol can be obtained with the command \diameter from the "wasysym" package.


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 metric. The page on
Jung's theorem In geometry, Jung's theorem is an inequality between the diameter of a set of points in any Euclidean space and the radius of the minimum enclosing ball of that set. It is named after Heinrich Jung, who first studied this inequality in 1901. Alg ...
discusses some more general inequalities relating the diameter to the radius.


See also

* * Caliper,
micrometer Micrometer can mean: * Micrometer (device), used for accurate measurements by means of a calibrated screw * American spelling of micrometre The micrometre ( international spelling as used by the International Bureau of Weights and Measures; ...
, tools for measuring diameters * * , a concept in group theory * Eratosthenes, who calculated the diameter of the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...
around 240 BC. * * * * * * * The diameters of a screwthread *


References

{{Authority control Elementary geometry Length Circles