Diameter
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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 chords of the circle. The word "diameter" derives from Greek διάμετρος (diametros), "diagonal of a circle", from δια- (dia-), "across, through" + μέτρον (metron), "a measure"^[1]).

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 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 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 differential geometry, the diameter is an important global Riemannian invariant.

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

Diameter symbol

Sign ⌀ from Autocad drawing

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 ⌀ 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, 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.

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

References

External links

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