View source for Length

{{Short description|Measure of distance in physical space}}
{{About|a physical measurement}}
{{Redirect|Width||Width (disambiguation)}}
{{Redirect|Breadth|ship measurements|Breadth (nautical)}}
{{Use British English|date=March 2018}}
{{Infobox physical quantity
| name       = Length
| image      = Scale kilometres miles.svg
| caption    = The [[Metric system|metric]] length of one [[kilometre]] is equivalent to the [[imperial measurement]] of 0.62137&nbsp;[[mile]]s.
| unit       = [[metre]] (m)
| otherunits = see [[unit of length]]
| symbols    = {{mvar|l}}
| dimension  = <math>\mathsf{L}</math>
| extensive  = yes
}}

'''Length''' is a measure of [[distance]]. In the [[International System of Quantities]], length is a [[quantity]] with [[Dimension (physical quantity)|dimension]] distance. In most [[systems of measurement]] a [[Base unit (measurement)|base unit]] for length is chosen, from which all other units are derived. In the [[International System of Units]] (SI) system the base unit for length is the [[metre]].<ref name=":0" />
 
Length is commonly understood to mean the most extended [[size|dimension]] of a fixed object.<ref name=":0">{{Cite encyclopedia|url=http://wordnetweb.princeton.edu/perl/webwn?s=length|encyclopedia=WordNet|title=Length|access-date=15 March 2020|archive-url=https://web.archive.org/web/20160925172753/http://wordnetweb.princeton.edu/perl/webwn?s=LENGTH|archive-date=25 September 2016|url-status=live}}</ref> However, this is not always the case and may depend on the position the object is in.

Various terms for the length of a fixed object are used, and these include [[height]], which is vertical length or vertical extent, width, breadth, and depth. ''Height'' is used when there is a base from which vertical measurements can be taken. ''Width'' and ''breadth'' usually refer to a shorter dimension than ''length''. ''Depth'' is used for the measure of a [[third dimension]].<ref>{{Cite web|url=http://thinkmath.edc.org/resource/measurement-length-width-height-depth|title=Measurement: Length, width, height, depth|website=Think Math!|access-date=15 March 2020|archive-url=https://web.archive.org/web/20200224001255/http://thinkmath.edc.org/resource/measurement-length-width-height-depth|archive-date=24 February 2020|url-status=live}}</ref>

Length is the measure of one spatial dimension, whereas [[area]] is a measure of two dimensions (length squared) and [[volume]] is a measure of three dimensions (length cubed).

== History ==
Measurement has been important ever since humans settled from nomadic lifestyles and started using building materials, occupying land and trading with neighbours. As trade between different places increased, the need for standard units of length increased. And later, as society has become more technologically oriented, much higher accuracy of measurement is required in an increasingly diverse set of fields, from micro-electronics to interplanetary ranging.<ref>History of Length Measurement, [http://resource.npl.co.uk/docs/educate_explore/posters/bg_historyoflength_poster.pdf  National Physical Laboratory] {{Webarchive|url=https://web.archive.org/web/20131126043209/http://resource.npl.co.uk/docs/educate_explore/posters/bg_historyoflength_poster.pdf |date=2013-11-26 }}</ref>

Under [[Albert Einstein|Einstein]]'s [[special relativity]], length can no longer be thought of as being constant in all [[reference frame]]s. Thus a [[Ruler (tool)|ruler]] that is one metre long in one frame of reference will not be one metre long in a reference frame that is moving relative to the first frame. This means the length of an object varies depending on the speed of the observer.

== Use in mathematics ==
=== Euclidean geometry ===
{{main|Euclidean geometry}}
In Euclidean geometry, length is measured along [[straight line]]s unless otherwise specified and refers to [[line segment|segments]] on them. [[Pythagorean theorem|Pythagoras's theorem]] relating the length of the sides of a [[right triangle]] is one of many applications in Euclidean geometry. Length may also be measured along other types of curves and is referred to as [[arclength]].

In a [[triangle]], the length of an [[Altitude (triangle)|altitude]], a line segment drawn from a vertex [[perpendicular]] to the side not passing through the vertex (referred to as a [[Base (geometry)|base]] of the triangle), is called the height of the triangle.

The [[area]] of a [[rectangle]] is defined to be length&thinsp;×&thinsp;width of the rectangle. If a long thin rectangle is stood up on its short side then its area could also be described as its height&thinsp;×&thinsp;width.

The [[volume]] of a [[Rectangular cuboid|solid rectangular box]] (such as a [[plank of wood]]) is often described as length&thinsp;×&thinsp;height&thinsp;×&thinsp;depth.

The [[perimeter]] of a [[polygon]] is the sum of the lengths of its [[Edge (geometry)|sides]].

The [[circumference]] of a circular [[Disk (mathematics)|disk]] is the length of the [[Boundary (of a manifold)|boundary]] (a [[Circle (geometry)|circle]]) of that disk.

=== Other geometries ===
{{Further|Non-Euclidean geometry}}
In other geometries, length may be measured along possibly curved paths, called [[geodesic]]s. The [[Riemannian geometry]] used in [[general relativity]] is an example of such a geometry. In [[spherical geometry]], length is measured along the [[great circles]] on the sphere and the distance between two points on the sphere is the shorter of the two lengths on the great circle, which is determined by the plane through the two points and the center of the sphere.

=== Graph theory ===
In an [[unweighted graph]], the length of a [[Cycle (graph theory)|cycle]], [[Path (graph theory)|path]], or [[Walk (graph theory)|walk]] is the number of [[Edge (graph theory)|edge]]s it uses.<ref>{{Cite web|url=https://primes.utm.edu/graph/glossary.html|title=Graph Theory Glossary|last=Caldwell|first=Chris K.|date=1995}}</ref> In a [[weighted graph]], it may instead be the sum of the weights of the edges that it uses.<ref>{{Cite web|url=http://www.mathcs.emory.edu/~cheung/Courses/323/Syllabus/Graph/dijkstra1.html|title=Weighted graphs and path length|last=Cheung|first=Shun Yan}}</ref>

Length is used to define the [[shortest path]], [[girth (graph theory)|girth]] (shortest cycle length), and [[longest path]] between two [[Vertex (graph theory)|vertices]] in a graph.

=== Measure theory ===
{{main|Lebesgue measure}}
In measure theory, length is most often generalized to general sets of <math>\mathbb{R}^n</math> via the [[Lebesgue measure]]. In the one-dimensional case, the Lebesgue outer measure of a set is defined in terms of the lengths of open intervals. Concretely, the length of an [[Open Interval|open interval]] is first defined as
: <math>\ell(\{x\in\mathbb R\mid a<x<b\})=b-a.</math>
so that the Lebesgue outer measure <math>\mu^*(E)</math> of a general set <math>E</math> may then be defined as<ref>{{cite web|url=http://zeta.math.utsa.edu/~mqr328/class/real2/L-measure.pdf|title=Lebesgue Measure|last=Le|first=Dung|url-status=live|archive-url=https://web.archive.org/web/20101130171814/http://zeta.math.utsa.edu/~mqr328/class/real2/L-measure.pdf|archive-date=2010-11-30}}</ref>
: <math>\mu^*(E)=\inf\left\{\sum_k \ell(I_k):I_k\text{ is a sequence of open intervals such that }E\subseteq\bigcup_k I_k\right\}.</math>

== Units ==
{{Main|Unit of length}}
In the physical sciences and engineering, when one speaks of {{em|[[unit of length|units of length]]}}, the word {{em|length}} is synonymous with [[distance]]. There are several [[Units of measurement|units]] that are used to [[Measurement|measure]] length. Historically, units of length may have been derived from the lengths of human body parts, the distance travelled in a number of paces, the distance between landmarks or places on the Earth, or arbitrarily on the length of some common object.

In the [[International System of Units]] (SI), the [[SI base unit|base unit]] of length is the [[metre]] (symbol, m), now defined in terms of the [[speed of light]] (about 300 million metres per [[second]]). The [[millimetre]] (mm), [[centimetre]] (cm) and the [[kilometre]] (km), derived from the metre, are also commonly used units. In [[U.S. customary units]], English or [[imperial system of units]], commonly used units of length are the [[inch]] (in), the [[foot (length)|foot]] (ft), the [[yard]] (yd), and the [[statute mile|mile]] (mi). A unit of length used in [[navigation]] is the [[nautical mile]] (nmi).<ref>{{cite book|last=Cardarelli|first=François|title=Encyclopaedia of Scientific Units, Weights, and Measures: Their SI Equivalences and Origins|url=https://archive.org/details/encyclopaediaofs0000card|url-access=registration|year=2003|publisher=Springer|isbn=9781852336820 }}</ref>

{{calculator|id=km|type=number|size=9|default=1.609344|formula=miles*1.609344}} km = 
{{calculator|id=miles|type=number|size=16|default=1|formula=km/1.609344}} miles 

Units used to denote distances in the vastness of space, as in [[astronomy]], are much longer than those typically used on Earth (metre or kilometre) and include the [[astronomical unit]] (au), the [[light-year]], and the [[parsec]] (pc).

Units used to denote sub-atomic distances, as in [[nuclear physics]], are much smaller than the millimetre. Examples include the [[fermi (unit)|fermi]] (fm).

== See also ==
* [[Arc length]]
* [[List of humorous units of measurement#Length|Humorous units of length]]
* [[Length measurement]]
* [[Metric system]]
* [[Metric units#Length|Metric units]]
* [[Orders of magnitude (length)]]
* [[Reciprocal length]]

== References ==
{{Wiktionary|length|distance|width|breadth}}
{{Commons category}}
{{Reflist}}
{{SI base quantities}}
{{Authority control}}

[[Category:Length| ]]
[[Category:Physical quantities]]
[[Category:SI base quantities]]