# Length

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Measure of distance in physical space

This article is about a physical measurement. For other uses, see [Length (disambiguation)](/source/Length_(disambiguation)).

"Width" redirects here. For other uses, see [Width (disambiguation)](/source/Width_(disambiguation)).

"Breadth" redirects here. For ship measurements, see [Breadth (nautical)](/source/Breadth_(nautical)).

Length The metric length of one kilometre is equivalent to the imperial measurement of 0.62137 miles. Common symbols l SI unit metre (m) Other units see unit of length Extensive? yes Dimension L {\displaystyle {\mathsf {L}}}

**Length** is a measure of [distance](/source/Distance). In the [International System of Quantities](/source/International_System_of_Quantities), length is a [quantity](/source/Quantity) with [dimension](/source/Dimension_(physical_quantity)) distance. In most [systems of measurement](/source/Systems_of_measurement) a [base unit](/source/Base_unit_(measurement)) for length is chosen, from which all other units are derived. In the [International System of Units](/source/International_System_of_Units) (SI) system, the base unit for length is the [metre](/source/Metre).[1]

Length is commonly understood to mean the most extended [dimension](/source/Size) of a fixed object.[1] 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](/source/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](/source/Third_dimension).[2]

Length is the measure of one spatial dimension, whereas [area](/source/Area) is a measure of two dimensions (length squared) and [volume](/source/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.[3]

Under [Einstein](/source/Albert_Einstein)'s [special relativity](/source/Special_relativity), length can no longer be thought of as being constant in all [reference frames](/source/Reference_frame). Thus a [ruler](/source/Ruler_(tool)) 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 article: [Euclidean geometry](/source/Euclidean_geometry)

In Euclidean geometry, length is measured along [straight lines](/source/Straight_line) unless otherwise specified and refers to [segments](/source/Line_segment) on them. [Pythagoras's theorem](/source/Pythagorean_theorem) relating the length of the sides of a [right triangle](/source/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](/source/Arclength).

In a [triangle](/source/Triangle), the length of an [altitude](/source/Altitude_(triangle)), a line segment drawn from a vertex [perpendicular](/source/Perpendicular) to the side not passing through the vertex (referred to as a [base](/source/Base_(geometry)) of the triangle), is called the height of the triangle.

The [area](/source/Area) of a [rectangle](/source/Rectangle) is defined to be length × 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 × width.

The [volume](/source/Volume) of a [solid rectangular box](/source/Rectangular_cuboid) (such as a [plank of wood](/source/Plank_of_wood)) is often described as length × height × depth.

The [perimeter](/source/Perimeter) of a [polygon](/source/Polygon) is the sum of the lengths of its [sides](/source/Edge_(geometry)).

The [circumference](/source/Circumference) of a circular [disk](/source/Disk_(mathematics)) is the length of the [boundary](/source/Boundary_(of_a_manifold)) (a [circle](/source/Circle_(geometry))) of that disk.

### Other geometries

Further information: [Non-Euclidean geometry](/source/Non-Euclidean_geometry)

In other geometries, length may be measured along possibly curved paths, called [geodesics](/source/Geodesic). The [Riemannian geometry](/source/Riemannian_geometry) used in [general relativity](/source/General_relativity) is an example of such a geometry. In [spherical geometry](/source/Spherical_geometry), length is measured along the [great circles](/source/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](/source/Unweighted_graph), the length of a [cycle](/source/Cycle_(graph_theory)), [path](/source/Path_(graph_theory)), or [walk](/source/Walk_(graph_theory)) is the number of [edges](/source/Edge_(graph_theory)) it uses.[4] In a [weighted graph](/source/Weighted_graph), it may instead be the sum of the weights of the edges that it uses.[5]

Length is used to define the [shortest path](/source/Shortest_path), [girth](/source/Girth_(graph_theory)) (shortest cycle length), and [longest path](/source/Longest_path) between two [vertices](/source/Vertex_(graph_theory)) in a graph.

### Measure theory

Main article: [Lebesgue measure](/source/Lebesgue_measure)

In measure theory, length is most often generalised to general sets of R n {\displaystyle \mathbb {R} ^{n}} via the [Lebesgue measure](/source/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](/source/Open_Interval) is first defined as

- ℓ ( { x ∈ R ∣ a < x < b } ) = b − a . {\displaystyle \ell (\{x\in \mathbb {R} \mid a<x<b\})=b-a.}

so that the Lebesgue outer measure μ ∗ ( E ) {\displaystyle \mu ^{*}(E)} of a general set E {\displaystyle E} may then be defined as[6]

- μ ∗ ( E ) = inf { ∑ k ℓ ( I k ) : I k is a sequence of open intervals such that E ⊆ ⋃ k I k } . {\displaystyle \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\}.}

### Computer science

Main article: [String (computer science)](/source/String_(computer_science))

In [computer science](/source/Computer_science), the length of a [string](/source/String_(computer_science)) is the number of [characters](/source/Character_(computing)) that comprise the string. For example, the [bit string](/source/Bit_array) 10010100001 {\displaystyle {\texttt {10010100001}}} has length 11.[7]

Many [programming languages](/source/Programming_language) have a string data type with a method for [length](/source/Comparison_of_programming_languages_(string_functions)#length) to retrieve the length of a string.

String s = "Hello, world!";
int len = s.length(); // len = 13

The "length" of a [sequential data structure](/source/Sequence_container_(C%2B%2B)), such as a [dynamic array](/source/Dynamic_array), can be described as the number of elements contained in it.

import java.util.List;

int[] a = {1, 2, 3, 4, 5};
int len = a.length; // len = 5

List<Integer> list = List.of(1, 2, 3, 4, 5, 6, 7, 8);
int len = list.size(); // len = 8

## Units

Main article: [Unit of length](/source/Unit_of_length)

In the physical sciences and engineering, when one speaks of *[units of length](/source/Unit_of_length)*, the word *length* is synonymous with [distance](/source/Distance). There are several [units](/source/Units_of_measurement) that are used to [measure](/source/Measurement) 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](/source/International_System_of_Units) (SI), the [base unit](/source/SI_base_unit) of length is the [metre](/source/Metre) (symbol, m), now defined in terms of the [speed of light](/source/Speed_of_light) (about 300 million metres per [second](/source/Second)). The [millimetre](/source/Millimetre) (mm), [centimetre](/source/Centimetre) (cm) and the [kilometre](/source/Kilometre) (km), derived from the metre, are also commonly used units. In [U.S. customary units](/source/U.S._customary_units), English or [imperial system of units](/source/Imperial_system_of_units), commonly used units of length are the [inch](/source/Inch) (in), the [foot](/source/Foot_(length)) (ft), the [yard](/source/Yard) (yd), and the [mile](/source/Statute_mile) (mi). A unit of length used in [navigation](/source/Navigation) is the [nautical mile](/source/Nautical_mile) (nmi).[8]

1.609344 km = 1 miles

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

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

## See also

- [Arc length](/source/Arc_length)

- [Humorous units of length](/source/List_of_humorous_units_of_measurement#Length)

- [Length measurement](/source/Length_measurement)

- [Metric system](/source/Metric_system)

- [Metric units](/source/Metric_units#Length)

- [Orders of magnitude (length)](/source/Orders_of_magnitude_(length))

- [Reciprocal length](/source/Reciprocal_length)

## References

Look up ***[length](https://en.wiktionary.org/wiki/length)***, ***[distance](https://en.wiktionary.org/wiki/distance)***, ***[width](https://en.wiktionary.org/wiki/width)***, or ***[breadth](https://en.wiktionary.org/wiki/breadth)*** in Wiktionary, the free dictionary.

Wikimedia Commons has media related to [Length](https://commons.wikimedia.org/wiki/Category:Length).

1. ^ [***a***](#cite_ref-:0_1-0) [***b***](#cite_ref-:0_1-1) ["Length"](http://wordnetweb.princeton.edu/perl/webwn?s=length). *WordNet*. [Archived](https://web.archive.org/web/20160925172753/http://wordnetweb.princeton.edu/perl/webwn?s=LENGTH) from the original on 25 September 2016. Retrieved 15 March 2020.

1. **[^](#cite_ref-2)** ["Measurement: Length, width, height, depth"](http://thinkmath.edc.org/resource/measurement-length-width-height-depth). *Think Math!*. [Archived](https://web.archive.org/web/20200224001255/http://thinkmath.edc.org/resource/measurement-length-width-height-depth) from the original on 24 February 2020. Retrieved 15 March 2020.

1. **[^](#cite_ref-3)** History of Length Measurement, [National Physical Laboratory](http://resource.npl.co.uk/docs/educate_explore/posters/bg_historyoflength_poster.pdf) [Archived](https://web.archive.org/web/20131126043209/http://resource.npl.co.uk/docs/educate_explore/posters/bg_historyoflength_poster.pdf) 2013-11-26 at the [Wayback Machine](/source/Wayback_Machine)

1. **[^](#cite_ref-4)** Caldwell, Chris K. (1995). ["Graph Theory Glossary"](https://web.archive.org/web/20221022091553/https://primes.utm.edu/graph/glossary.html). Archived from [the original](https://primes.utm.edu/graph/glossary.html) on 2022-10-22. Retrieved 2020-03-18.

1. **[^](#cite_ref-5)** Cheung, Shun Yan. ["Weighted graphs and path length"](https://web.archive.org/web/20200109145000/http://www.mathcs.emory.edu/~cheung/Courses/323/Syllabus/Graph/dijkstra1.html). Archived from [the original](http://www.mathcs.emory.edu/~cheung/Courses/323/Syllabus/Graph/dijkstra1.html) on 2020-01-09. Retrieved 2020-03-18.

1. **[^](#cite_ref-6)** Le, Dung. ["Lebesgue Measure"](http://zeta.math.utsa.edu/~mqr328/class/real2/L-measure.pdf) (PDF). [Archived](https://web.archive.org/web/20101130171814/http://zeta.math.utsa.edu/~mqr328/class/real2/L-measure.pdf) (PDF) from the original on 2010-11-30.

1. **[^](#cite_ref-7)** John E. Hopcroft, Jeffrey D. Ullman (1979). [*Introduction to Automata Theory, Languages, and Computation*](https://archive.org/details/introductiontoau00hopc). Addison-Wesley. [ISBN](/source/ISBN_(identifier)) [0-201-02988-X](https://en.wikipedia.org/wiki/Special:BookSources/0-201-02988-X). Here: sect.1.1, p.1

1. **[^](#cite_ref-8)** Cardarelli, François (2003). [*Encyclopaedia of Scientific Units, Weights, and Measures: Their SI Equivalences and Origins*](https://archive.org/details/encyclopaediaofs0000card). Springer. [ISBN](/source/ISBN_(identifier)) [9781852336820](https://en.wikipedia.org/wiki/Special:BookSources/9781852336820).

v t e SI base quantities Base quantities Quantity SI unit Name Symbol Dimension symbol Unit name Unit symbol time, duration t T second s length l, x, r, etc. L metre m mass m M kilogram kg electric current I , i I ampere A thermodynamic temperature T Θ kelvin K amount of substance n N mole mol luminous intensity Iv J candela cd See also History of the metric system International System of Quantities 2019 revision Systems of measurement Category Outline

Authority control databases International GND Other Yale LUX

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