length

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OR:

Length is a measure of
distance Distance is a numerical or occasionally qualitative 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"). ...
. In the
International System of Quantities The International System of Quantities (ISQ) consists of the Quantity, quantities used in physics and in modern science in general, starting with basic quantities such as length and mass, and the relationships between those quantities. This syste ...
, length is a
quantity Quantity or amount 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 multiple of a u ...
with
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
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 i ...
a base unit for length is chosen, from which all other units are derived. In the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
(SI) system the base unit for length is the
metre The metre (British English, British spelling) or meter (American English, American spelling; American and British English spelling differences#-re, -er, see spelling differences) (from the French unit , from the Greek language, Greek noun , "m ...
. Length is commonly understood to mean the most extended
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
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 Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...
, 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 region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an o ...
is a measure of two dimensions (length squared) and
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
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 physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...
's
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...
, length can no longer be thought of as being constant in all
reference frame In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin (mathematics), origin, orientation (geometry), orientation, and scale (geometry), scale are specified by a set of reference point ...
s. Thus a
ruler A ruler, sometimes called a rule, line gauge, or scale, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to distance measurement, measure distances or draw straight lines. Variant ...
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 In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, ...
s unless otherwise specified and refers to segments on them. Pythagoras's theorem relating the length of the sides of a
right triangle A right triangle (American English) or right-angled triangle (British English, British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one an ...
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 A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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, an ...
, 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 Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works i ...
to the side not passing through the vertex (referred to as a base of the triangle), is called the height of the triangle. The
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an o ...
of a
rectangle In Euclidean geometry, 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 par ...
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 a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of a solid rectangular box (such as a plank of wood) is often described as length × height × depth. The
perimeter A perimeter is a closed path (geometry), path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length (mathematics), length. The perimeter of a circle or an ellipse is called its circumference. Calc ...
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 bounded plane region, the bounding circuit, or the tw ...
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 ...
of a circular disk is the length of the boundary (a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. Equivalently, it is the curve traced out by a point that moves in ...
) of that disk.

## Other geometries

In other geometries, length may be measured along possibly curved paths, called
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path (arc (geometry), arc) between two points in a differential geometry of surfaces, surface, or more generally in a Riemannian manifold. The term also has meaning in ...
s. The
Riemannian geometry Riemannian geometry is the branch of differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniq ...
used in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
is an example of such a geometry. In
spherical geometry file:Spherical_triangle_3d.png, 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other ...
, 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, path, or
walk Walking (also known as ambulation) is one of the main gait Gait is the pattern of Motion (physics), movement of the limb (anatomy), limbs of animals, including Gait (human), humans, during Animal locomotion, locomotion over a solid substrat ...
is the number of
edge Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by ...
s it uses. In a
weighted graph This is a glossary of graph theory. Graph theory is the study of graph (discrete mathematics), graphs, systems of nodes or vertex (graph theory), vertices connected in pairs by lines or #edge, edges. Symbols A ...
, 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 (graph theory), path between two vertex (graph theory), vertices (or nodes) in a Graph (discrete mathematics), graph such that the sum of the Glossary of graph theory te ...
, girth (shortest cycle length), and
longest path In graph theory and theoretical computer science, the longest path problem is the problem of finding a simple path (graph theory), path of maximum length in a given Graph (discrete mathematics), graph. A path is called ''simple'' if it does not ha ...
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 measure In Measure (mathematics), measure theory, a branch of mathematics, the Lebesgue measure, named after france, French mathematician Henri Lebesgue, is the standard way of assigning a measure (mathematics), measure to subsets of ''n''-dimensional Eucl ...
. 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, a (real) interval is a set (mathematics), 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 ...
is first defined as :$\ell\left(\\right)=b-a.$ so that the Lebesgue outer measure $\mu^*\left(E\right)$ of a general set $E$ may then be defined as :$\mu^*\left(E\right)=\inf\left\.$

# Units

In the physical sciences and engineering, when one speaks of , the word is synonymous with
distance Distance is a numerical or occasionally qualitative 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"). ...
. There are several units 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 The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
(SI), the base unit of length is the
metre The metre (British English, British spelling) or meter (American English, American spelling; American and British English spelling differences#-re, -er, see spelling differences) (from the French unit , from the Greek language, Greek noun , "m ...
(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 that is important in many areas of physics. The speed of light is exactly equal to ). According to the special relativity, special theory of relativity, is ...
second The second (symbol: s) is the unit of Time in physics, time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally t ...
). The
millimetre file:EM Spectrum Properties edit.svg, 330px, Different lengths as in respect to the electromagnetic spectrum, measured by the metre and its derived scales. The microwave is between 1 meter to 1 millimeter. The millimetre (American and British Eng ...
(mm),
centimetre file:EM Spectrum Properties edit.svg, 330px, Different lengths as in respect to the Electromagnetic spectrum, measured by the Metre and its deriveds scales. The Microwave are in-between 1 meter to 1 millimeter. A centimetre (international spell ...
(cm) and the
kilometre The kilometre (SI symbol: km; or ), spelt kilometer in American English, is a unit of length in the International System of Units (SI), equal to one thousand metres (kilo- being the SI prefix for ). It is now the measurement unit used for exp ...
(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 Measuring tape with inches The inch (symbol: in or ″) is a unit of length in the British imperial and the United States customary systems of measurement. It is equal to yard or of a foot. Derived from the Roman uncia ("twelf ...
(in), the
foot The foot (plural, : feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a limb (anatomy), limb which bears weight and allows animal locomotion, locomotion. In many animals with feet, the foot is a separate o ...
(ft), the
yard The yard (symbol: yd) is an English units, English unit of length in both the British imperial units, imperial and US United States customary units, customary systems of measurement equalling 3 foot (unit), feet or 36 inches. Sinc ...
(yd), and the
mile The mile, sometimes the international mile or statute mile to distinguish it from other miles, is a imperial unit, British imperial unit and United States customary unit of distance; both are based on the older English unit of Unit of length, ...
(mi). A unit of length used in
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the motion, movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navi ...
is the
nautical mile A nautical mile is a unit of length used in air, marine, and space navigation, and for the definition of territorial waters. Historically, it was defined as the meridian arc length corresponding to one minute of arc, minute ( of a degree) of la ...
(nmi). Units used to denote distances in the vastness of space, as in
astronomy Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
, 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 Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large list of largest lakes and ...
(au), the
light-year A light-year, alternatively spelled light year, is a large unit of length used to express astronomical distances and is equivalent to about 9.46  trillion kilometers (), or 5.88 trillion miles ().One trillion here is taken to be 101 ...
, 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 (au), i.e. . The parsec unit is obtained by the use of parallax and trigonometry, and ...
(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, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies the ...
, are much smaller than the centimetre. Examples include 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 Atomic Age, nuclea ...
.

*
Arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
*
Conversion of units Conversion of units is the conversion between different units of measurement A unit of measurement is a definite magnitude (mathematics), magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for ...
* Humorous units of length * Length measurement *
Metric system The metric system is a system of measurement that succeeded the Decimal, decimalised system based on the metre that had been introduced in French Revolution, France in the 1790s. The historical development of these systems culminated in the d ...
*
Metric units Metric units are unit of measurement, 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 the ...
*
Orders of magnitude (length) The following are examples of orders of magnitude An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the r ...
*
Reciprocal length Reciprocal length or inverse length is a quantity Quantity or amount 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", ...

# References

{{Authority control Physical quantities SI base quantities