# Diameter

Circle with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta.

In geometry, 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.

In more modern usage, the length of a diameter is also called the diameter. In this sense one speaks of the diameter rather than a diameter (which refers to the line itself), because all diameters of a circle or sphere have the same length, this being twice the radius r.

${\displaystyle d=2r\quad \Rightarrow \quad r={\frac {d}{2}}.}$

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 width is often defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers.[1] For a 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.

For an ellipse, the standard terminology is different. A diameter of an ellipse is any chord passing through the center of the ellipse.[2] For example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one of them is parallel to the other one. The longest diameter is called the major axis.

The word "diameter" is derived from Greek διάμετρος (diametros), "diameter of a circle", from διά (dia), "across, through" and μέτρον (metron), "measure".[3] It is often abbreviated DIA, dia, d, or .

## 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 of scattered points. The diameter 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. So, if A is the subset, the diameter is

sup { d(x, y) | x, yA } .

If the distance function d is viewed here as having codomain R (the set of all real numbers), this implies that the diameter of the empty set (the case A = ∅) equals −∞ (negative infinity). Some authors prefer to treat the empty set as a special case, assigning it a diameter equal to 0,[4] 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, 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 supremum 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 plane 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.

Other Languages
Afrikaans: Deursnee
Alemannisch: Durchmesser
العربية: قطر (هندسة)
asturianu: Diámetru
azərbaycanca: Diametr
বাংলা: ব্যাস
Bân-lâm-gú: Ti̍t-kèng
беларуская: Дыяметр
беларуская (тарашкевіца)‎: Дыямэтар
български: Диаметър
bosanski: Prečnik
català: Diàmetre
Чӑвашла: Диаметр
chiShona: Gurapakati
Cymraeg: Diamedr
dansk: Diameter
Deutsch: Durchmesser
eesti: Diameeter
Ελληνικά: Διάμετρος
español: Diámetro
Esperanto: Diametro
euskara: Diametro
français: Diamètre
galego: Diámetro
한국어: 지름
հայերեն: Տրամագիծ
hrvatski: Promjer
Bahasa Indonesia: Diameter
íslenska: Þvermál
italiano: Diametro
עברית: קוטר
Basa Jawa: Dhiamèter
ქართული: დიამეტრი
қазақша: Диаметр
Kiswahili: Kipenyo
Latina: Diametros
latviešu: Diametrs
Lëtzebuergesch: Duerchmiesser
lietuvių: Skersmuo
Limburgs: Diameter
македонски: Пречник
മലയാളം: വ്യാസം
Bahasa Melayu: Diameter
မြန်မာဘာသာ: အချင်း
Nederlands: Diameter

Nordfriisk: Trochmeeder
norsk: Diameter
norsk nynorsk: Diameter
occitan: Diamètre
oʻzbekcha/ўзбекча: Diametr
ភាសាខ្មែរ: អង្កត់ផ្ចិត
Plattdüütsch: Dörmeter
polski: Średnica
português: Diâmetro
română: Diametru
Runa Simi: Raqta
русский: Диаметр
Scots: Diameter
Simple English: Diameter
slovenčina: Priemer (geometria)
slovenščina: Premer
ślůnski: Szyrzka
Soomaaliga: Dhexroor
српски / srpski: Пречник
srpskohrvatski / српскохрватски: Promjer
suomi: Halkaisija
svenska: Diameter
Tagalog: Diyametro
தமிழ்: விட்டம்
татарча/tatarça: Диаметр
тоҷикӣ: Диаметр
Türkçe: Çap
українська: Діаметр
Tiếng Việt: Đường kính
Winaray: Diametro
ייִדיש: דיאמעטער