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Diameter
In geometry, a '''diameter''' of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chord (geometry)|chords of the circle. The word "diameter" derives from Greek language|Greek ''διάμετρος'' (''diametros''), "diagonal of a circle", from ''δια-'' (''dia-''), "across, through" + ''μέτρον'' (''metron''), "a measure"[Online Etymology Dictionary]).
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, because all diameters of a circle have the same length, this being twice the radius.
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 defined to be the smallest such distance. 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. See also Tangent lines to circles.
The '''diameter''' of a connected graph is the distance between the two vertex (graph theory)|vertices which are furthest from each other. The distance between two vertices ''a'' and ''b'' is the length of the shortest path connecting them (for the length of a path, see Graph theory).
Generalisations
The three definitions given above are special cases of a more general definition. The '''diameter''' of a subset of a metric space is the supremum|least upper bound of the distances between pairs of points in the subset. So, if ''A'' is the subset, the diameter is
- supremum|sup { d(''x'', ''y'') | ''x'', ''y'' ∈ ''A'' } .
In differential geometry, the diameter is an important global Riemannian geometry|Riemannian invariant.
In medical Idiom#Parlance|parlance the diameter of a lesion is the longest line segment whose endpoints are within the lesion.
Diameter symbol
The symbol or variable for diameter is similar in size and design to ø, the lowercase letter o with stroke. Unicode provides character number 8960 (hexadecimal 2300) for the symbol, which can be encoded in HTML webpages as ⌀ or ⌀. Proper display of this character, however, is unlikely in most situations, as most Typeface|fonts do not have it included. (Your browser displays ⌀ in the current font.) In most situations the letter ø is acceptable, obtained in Microsoft Windows by holding the Alt key down while entering 0 2 4 8 on the numeric keypad.
The diameter symbol, , is distinct from the empty set symbol, ∅, from an uppercase Phi (letter)|phi, Φ, and the Nordic vowel, Ø.
The diameter also refers to the approximate size of the corner of a frame of any given object to the nearest flat surface it represents.
Graph theory
To find diameter of a graph (mathematics)|graph, first find the shortest path (graph theory)|path between each pair of vertex|vertices. The longest of these paths is the diameter of the graph.
See also
- angular diameter
- hydraulic diameter
- caliper, micrometer, tools for measuring diameters
- Eratosthenes, who calculated the diameter of the Earth around 240 BC.
- Jung's theorem, an inequality relating the diameter to the radius of the circumradius|smallest enclosing ball
References
External links
- (geometry) Diameter and many other circle properties defined With interactive applets
Category:Elementary geometry
Category:Length
Category:Greek loanwords
simple:Diameter
zh-yue:直徑
Related Images- Diameter
Sources: StartLearningNow, Wikipedia | Usage license: GNU FDL
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