Diameter

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For the authentication, authorisation, and accounting protocol, see DIAMETER.

In geometry, a diameter (Greek words diairo = divide and metro = measure) of a circle is any straight line segment that passes through the centre and whose endpoints are on the circular boundary, or, in more modern usage, the length of such a line segment. When using the word in the more modern sense, one speaks of the diameter rather than a diameter, because all diameters of a circle have the same length. This length is twice the radius. The diameter of a circle is also the longest chord that the circle has.

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.

The diameter of a connected graph is the distance between the two 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).

The three definitions given above are special cases of a more general definition. The diameter of a subset of a metric space is the least upper bound of the distances between pairs of points in the subset. So, if A is the subset, the diameter is

sup { d(x, y) | x, y ∈ A } .

In medical parlance the diameter of a lesion is the longest line segment whose endpoints are within the lesion.

[edit] 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 fonts do not have it included. (Your browser displays ⌀ and ⌀ 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.

It is important not to confuse a diameter symbol (ø) with the empty set symbol, similar to the uppercase Ø. Phi is sometimes used for diameter, although this seems to come from the fact that the symbols appear similar.

[edit] 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 smallest enclosing ball

[edit] External links

• (geometry) Diameter and many other circle properties defined With interactive applets