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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, 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 mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
distance. In most systems of measurement a 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 The metre ( British spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pre ...
. Length is commonly understood to mean the most extended
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
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 ab ...
, 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 op ...
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's
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
, 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, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
s. Thus a 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

In Euclidean geometry, length is measured along straight lines unless otherwise specified and refers to segments on them.
Pythagoras's theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between 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 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 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, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
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 op ...
of a 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 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 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 two ...
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 ...
of a circular disk is the length of the boundary (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 cons ...
) of that disk.


Other geometries

In other geometries, length may be measured along possibly curved paths, called geodesics. The Riemannian geometry used in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
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, path, or walk is the number of edges it uses. In a weighted graph, it may instead be the sum of the weights of the edges that it uses. Length is used to define the shortest path,
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 ...
(shortest cycle length), and longest path 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 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 wi ...
. 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 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 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 (SI), the base unit of length is the
metre The metre ( British spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pre ...
(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 theory of relativity, is the upper limit fo ...
(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 (ft), the yard (yd), and the mile (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 ...
is the nautical mile (nmi). Units used to denote distances in the vastness of space, as in
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
, 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 approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbits ...
(au), the light-year, and the parsec (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 t ...
, 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 nuclear age" and t ...
.


See also

*
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 * Humorous units of length *
Length measurement Length measurement, distance measurement, or range measurement (ranging) refers to the many ways in which length, distance, or range can be measured. The most commonly used approaches are the rulers, followed by transit-time methods and the int ...
*
Metric system The metric system is a system of measurement that succeeded the decimalised system based on the metre that had been introduced in France in the 1790s. The historical development of these systems culminated in the definition of the Intern ...
* Metric units * Orders of magnitude (length) * Reciprocal length


References

{{Authority control Physical quantities SI base quantities