geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a diameter of a

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

is any straight

line segment In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...

that passes through the centre 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 (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

. 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 In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments 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 th ...

r.

d = 2r \qquad\text\qquad r = \frac.

The word "diameter" is derived from (), "diameter of a circle", from (), "across, through" and (), "measure". It is often abbreviated

\text, \text, d,

\varnothing.

Constructions

With

straightedge and compass In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

, a diameter of a given circle can be constructed as the

perpendicular bisector In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''se ...

of an arbitrary chord. Drawing two diameters in this way can be used to locate the center of a circle, as their crossing point. 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 In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...

of the segment and then drawing the circle centered at the midpoint through one of the ends of the line segment.

Symbol

The

symbol A symbol is a mark, Sign (semiotics), sign, or word that indicates, signifies, or is understood as representing an idea, physical object, object, or wikt:relationship, relationship. Symbols allow people to go beyond what is known or seen by cr ...

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. Photographic

filter thread In photography and cinematography, a filter is a camera accessory consisting of an optical filter that can be inserted into the optical path. The filter can be of a square or oblong shape and mounted in a holder accessory, or, more commonly, a gl ...

sizes are often denoted in this way. The symbol has a

code point A code point, codepoint or code position is a particular position in a Table (database), table, where the position has been assigned a meaning. The table may be one dimensional (a column), two dimensional (like cells in a spreadsheet), three dime ...

Unicode Unicode or ''The Unicode Standard'' or TUS is a character encoding standard maintained by the Unicode Consortium designed to support the use of text in all of the world's writing systems that can be digitized. Version 16.0 defines 154,998 Char ...

at , in the

Miscellaneous Technical Miscellaneous Technical is a Unicode block ranging from U+2300 to U+23FF. It contains various common symbols which are related to and used in the various technical, programming language, and academic professions. For example: * Symbol ⌂ (HTML ...

set. It should not be confused with several other characters (such as or ) that resemble it but have unrelated meanings. It has the

compose sequence A compose key (sometimes called multi key) is a key on a computer keyboard that indicates that the following (usually 2 or more) keystrokes trigger the insertion of an alternate character, typically a precomposed character or a symbol. For insta ...

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

n

-dimensional object, 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 ''

diameter of a set In mathematics, the diameter of a set of points in a metric space is the largest distance between points in the set. As an important special case, the diameter of a metric space is the largest distance between any two points in the space. This ...

'' is the

least upper bound In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...

of the set of all distances between pairs of points in the subset. A different and incompatible definition is sometimes used for the diameter of a

conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...

. In this context, a diameter is any chord which passes through the conic's centre. A diameter of an

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

is any line passing through the centre of the ellipse. Half of any such diameter may be called a ''semidiameter'', although this term is most often a synonym for the

of a circle or sphere. The longest and shortest diameters are 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 longe ...

'' and ''minor axis'', respectively. ''

Conjugate diameters In geometry, two diameters of a conic section are said to be conjugate if each chord (geometry), chord parallel (geometry), parallel to one diameter is bisection, bisected by the other diameter. For example, two diameters of a circle are conjugate ...

'' 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 In applied sciences, the equivalent radius (or mean radius) is the radius of a circle or sphere with the same perimeter, area, or volume of a non-circular or non-spherical object. The equivalent diameter (or mean diameter) (D) is twice the equiva ...

'', the diameter of a circular or spherical approximation to the object. This includes

hydraulic diameter The hydraulic diameter, , is a commonly used term when handling flow in non-circular tubes and channels. Using this term, one can calculate many things in the same way as for a round tube. When the cross-section is uniform along the tube or channe ...

, the equivalent diameter of a channel or pipe through which liquid flows, and the

Sauter mean diameter In fluid dynamics, Sauter mean diameter (SMD) is an average measure of particle size. It was originally developed by German scientist Josef Sauter in the late 1920s. It is defined as the diameter of a sphere that has the same volume/surface area ...

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 metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is oc ...

Jung's theorem In geometry, Jung's theorem is an inequality (mathematics), inequality between the diameter of a set of points in any Euclidean space and the radius of the circumradius, minimum enclosing ball of that set. It is named after Heinrich Jung, who firs ...

provides more general inequalities relating the diameter to the radius.

References

{{Authority control Elementary geometry Length Circles

Constructions

Symbol

Generalizations

See also

References