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Length is a measure of
distance Distance is a numerical measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). The distance from a point A to ...
. In the
International System of Quantities The International System of Quantities (ISQ) is a set of quantities and the equations that relate them describing physics and nature, as used in modern science, officialized by the International Organization for Standardization (ISO) by year 2009. ...
, length is a
quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measureme ...
with
dimension thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to s ...
distance. In most
systems of measurement A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purposes of science and commerce. Systems of measurement in ...
a base unit for length is chosen, from which all other units are derived. In the
International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * ''International'' (Kevin Michael album), 2011 * ''International'' (New Order album), 2002 * ''International'' (The Three ...
(SI) system the base unit for length is the
metre The metre (Commonwealth spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure", and cognate with Sanskrit , meaning "measured") is the base unit of length in the Internation ...
. Length is commonly understood to mean the most extended
dimension thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to s ...
of a fixed object. 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 200px, A cuboid demonstrating the dimensions length, width">length.html" style="text-decoration: none;"class="mw-redirect" title="cuboid demonstrating the dimensions length">cuboid demonstrating the dimensions length, width, and height. H ...
, which is vertical length or vertical extent, and width, breadth or depth. Height is used when there is a base from which vertical measurements can be taken. Width or breadth usually refer to a shorter dimension when length is the longest one. Depth is used for the third dimension of a three dimensional object. Length is the measure of one spatial dimension, whereas
area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of ...

area
is a measure of two dimensions (length squared) and
volume Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using the SI derived unit ...
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. Under
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity, but he also ma ...
's
special relativity#REDIRECT Special relativity#REDIRECT Special relativity {{Redirect category shell, 1= {{R from other capitalisation ...
{{Redirect category shell, 1= {{R from other capitalisation ...
, length can no longer be thought of as being constant in all
reference frame In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements within that frame . ...
s. Thus a
ruler A ruler, sometimes called a rule or line gauge, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to measure distances or draw straight lines. Variants Rulers have long been made f ...
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

In Euclidean geometry, length is measured along
straight line 290px, A representation of one line segment. In geometry, the notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth. Lines are an idealiz ...
s unless otherwise specified and refers to segments on them.
Pythagoras's theorem In mathematics, the Pythagorean theorem, or Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
relating the length of the sides of a
right triangle A right triangle (American English) or right-angled triangle (British ) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. T ...
is one of many applications in Euclidean geometry. Length may also be measured along other types of curves and is referred to as
arclength Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment is also called of a curve. The advent of infinitesimal calculus led to a general formula that provides closed-form solu ...

arclength
. In a
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear, d ...
, the length of an
altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ( ...
, a line segment drawn from a vertex
perpendicular In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects. A line is said to be perpend ...
to the side not passing through the vertex (referred to as a
base Base or BASE may refer to: Brands and enterprises *Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM, an enterprise software company founded in 2009 with offices in Mountain View and Kraków, Poland *Base De ...
of the triangle), is called the height of the triangle. The
area Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of ...

area
of a
rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containing ...

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 Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using the SI derived unit ...
of a solid rectangular box (such as a plank of wood) is often described as length × height × depth. The
perimeter A perimeter is either a path that encompasses/surrounds/outlines a shape (in two dimensions) or its length (one-dimensional). The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical ...
of a
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain or ''polygonal circuit''. The solid plane region, the bounding circuit, or the two together, may ...

polygon
is the sum of the lengths of its sides. The
circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to a l ...
of a circular
disk Disk or disc may refer to: * Disk (mathematics) * Disk storage Music * Disc (band), an American experimental music band * ''Disk'' (album), a 1995 EP by Moby Other uses * Disc (galaxy), a disc-shaped group of stars * ''Disc'' (magazine), a Briti ...
is the length of the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *''Boundaries'' (2016 film), a 2016 Canadian film *''Boundaries'' (2018 film), a 2018 American-Canadian road trip film Mathematics and physics *Boundary (topolog ...
(a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. T ...
) of that disk.


Other geometries

In other geometries, length may be measured along possibly curved paths, called
geodesic In geometry, a geodesic () is commonly a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connec ...
s. The
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. ...
used in
general relativity General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes spec ...
is an example of such a geometry. In
spherical geometry Image:Triangles (spherical geometry).jpg, 300px, The sum of the angles of a spherical triangle is not equal to 180°. A sphere is a curved surface, but locally the laws of the flat (planar) Euclidean geometry are good approximations. In a small tr ...
, length is measured along the
great circles Great may refer to: Descriptions or measurements * Great, a relative measurement in physical space, see Size * Greatness, being divine, majestic, superior, majestic, or transcendent People with the name * "The Great", a historical suffix to people ...
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 Cycle or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social scienc ...
,
path#redirect Path {{R from ambiguous ...
, or
walk Walking (also known as ambulation) is one of the main gaits of terrestrial locomotion among legged animals. Walking is typically slower than running and other gaits. Walking is defined by an 'inverted pendulum' gait in which the body vaults over ...
is the number of
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{{Red ...
s it uses. In a
weighted graph This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B ...
, it may instead be the sum of the weights of the edges that it uses. Length is used to define the
shortest path In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two in ...
,
girth Girth may refer to: ;Mathematics * Girth (functional analysis), the length of the shortest centrally symmetric simple closed curve on the unit sphere of a Banach space * Girth (geometry), the perimeter of a parallel projection of a shape * Girth (g ...
(shortest cycle length), and
longest pathIn graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path of maximum length in a given graph. A path is called simple if it does not have any repeated vertices; the length of a path may either ...
between two vertices in a graph.


Measure theory

In measure theory, length is most often generalized to general sets of \mathbb^n via the
Lebesgue measureIn measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides with t ...
. 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 In mathematics, an (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
is first defined as :\ell(\)=b-a. so that the Lebesgue outer measure \mu^*(E) of a general set E may then be defined as :\mu^*(E)=\inf\left\.


Units

In the physical sciences and engineering, when one speaks of , the word is synonymous with
distance Distance is a numerical measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). The distance from a point A to ...
. There are several
units Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (album ...
that are used to measure length. Historically, units of length may have been derived from the lengths of human body parts, the distance traveled 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 International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * ''International'' (Kevin Michael album), 2011 * ''International'' (New Order album), 2002 * ''International'' (The Three ...
(SI), the
base unit
base unit
of length is the
metre The metre (Commonwealth spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure", and cognate with Sanskrit , meaning "measured") is the base unit of length in the Internation ...
(symbol, m) and is now defined in terms of the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant important in many areas of physics. Its exact value is defined as (approximately ). It is exact because, by international agreement, a metre is defined as the ...
(about 300 million metres per
second The second (symbol: s, abbreviation: sec) is the base unit of time in the International System of Units (SI) (French: Système International d’unités), commonly understood and historically defined as of a day – this factor derived from th ...
). The
millimetre The millimetre (international spelling; SI unit symbol mm) or millimeter (American spelling) is a unit of length in the metric system, equal to one thousandth of a metre, which is the SI base unit of length. Therefore, there are one thousand mill ...
(mm),
centimetre A centimetre (international spelling) or centimeter (American spelling) (SI symbol cm) is a unit of length in the metric system, equal to one hundredth of a metre, ''centi'' being the SI prefix for a factor of . The centimetre was the base unit ...
(cm) and the
kilometre The kilometre (SI symbol: km; or ), spelt kilometer in American English, is a unit of length in the metric system, equal to one thousand metres (kilo- being the SI prefix for ). It is now the measurement unit used for expressing distances betw ...
(km), derived from the metre, are also commonly used units. In
U.S. customary units United States customary units (U.S. customary units) are a system of measurements commonly used in the United States since it was formalized in 1832. The United States customary system (USCS or USC) developed from English units which were in use i ...
, English or
Imperial system of units The imperial system of units, imperial system or imperial units (also known as British Imperial or Exchequer Standards of 1826) is the system of units first defined in the British Weights and Measures Act 1824 and continued to be developed throu ...
, commonly used units of length are the
inch Measuring tape with inches The inch (symbol: in or ″) is a unit of length in the (British) imperial and United States customary systems of measurement. It is equal to yard or of a foot. Derived from the Roman uncia ("twelfth"), the word ...
(in), the
foot The foot (plural: feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a limb which bears weight and allows locomotion. In many animals with feet, the foot is a separate organ at the terminal part of the leg m ...
(ft), the
yard 300px, Bronze Yard №11, the official standard of length for the Treasury Department formally adopted a metric standard. Bronze Yard №11 was forged to be an exact copy of the British Imperial Standard Yard held by Parliament._Both_are_line_st ...
(yd), and the
mile The mile, sometimes the international mile to distinguish it from other miles, is an imperial unit and US customary unit (and previously an English unit) of length equal to 5,280 feet, or 1,760 yards, and standardised as exactly 1,609.344&nb ...
(mi). A unit of length used in
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, mar ...
is the
nautical mile#REDIRECT nautical mile#REDIRECT nautical mile {{Redirect category shell, 1= {{R from other capitalisation ...
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(nmi). Units used to denote distances in the vastness of space, as in
astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain t ...
, are much longer than those typically used on Earth (metre or centimetre) and include the
astronomical unit The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and equal to about or ~8 light minutes. The actual distance varies by about 3% as Earth orbits the Sun, from a maximum (aphelion ...
(au), the
light-year The light-year is a unit of length used to express astronomical distances and is equivalent to about 9.46 trillion kilometres () or 5.88 trillion miles ().One trillion here is taken to be 1012 (one million million, or billion in long sc ...
, and the
parsec The parsec (symbol: pc) is a unit of length used to measure the large distances to astronomical objects outside the Solar System, approximately equal to or , i. e. . Parsec is obtained by the use of parallax and trigonometry, and is defined as ...
(pc). Units used to denote sub-atomic distances, as in
nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied. Nuclear physics should not be confused with atomic physics, which studies the atom as a wh ...
, are much smaller than the centimetre. Examples include the
dalton Dalton may refer to: Science * Dalton (crater), a lunar crater * Dalton (program), chemistry software * Dalton (unit) (Da), the atomic mass unit Entertainment * Dalton (Buffyverse), minor character from ''Buffy the Vampire Slayer'' television s ...
and the
fermi Enrico Fermi (; 29 September 1901 - 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and the ...
.


See also

*
Conversion of units Conversion or convert may refer to: Arts, entertainment, and media * "Conversion" (''Doctor Who'' audio), an episode of the audio drama ''Cyberman'' * "Conversion" (''Stargate Atlantis''), an episode of the television series * "The Conversion" ('' ...
* Humorous units of length *
Metric system A metric system is a system of measurement that succeeded the decimalised system based on the metre introduced in France in the 1790s. The historical development of these systems culminated in the definition of the International System of Unit ...
*
Metric unitsMetric units are units based on the metre, gram or second and decimal (power of ten) multiples or sub-multiples of these. The most widely used examples are the units of the International System of Units (SI). By extension they include units of elec ...
*
Orders of magnitude (length) The following are examples of orders of magnitude for different lengths. __TOC__ Overview Detailed list To help compare different orders of magnitude, the following list describes various lengths between 1.6 \times 10^ metres and 10^ ...
*
Reciprocal lengthReciprocal length or inverse length is a measurement used in several branches of science and mathematics. As the reciprocal of length, common units used for this measurement include the reciprocal metre or inverse metre (symbol: m−1), the recip ...


References

{{Authority control Physical quantities
SI base quantities{{cat main, SI base unit Base quantities Physical quantities ...