Length

Length
SI unit		metre (m)
Other units	see unit of length
Extensive?	yes
Dimension	${\displaystyle {\mathsf {L}}}$

Length is a measure of

.

Length is commonly understood to mean the most extended dimension 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, 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.^[2]

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.^[3]

Under

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

.

In a triangle, the length of an altitude, a line segment drawn from a vertex perpendicular 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 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

) is often described as length × height × depth.

The perimeter of a polygon is the sum of the lengths of its sides.

The

) of that disk.

Other geometries

In other geometries, length may be measured along possibly curved paths, called

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

Length is used to define the

in a graph.

Measure theory

In measure theory, length is most often generalized to general sets of ${\displaystyle \mathbb {R} ^{n}}$ via the

is first defined as

${\displaystyle \ell (\{x\in \mathbb {R} \mid a<x<b\})=b-a.}$

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

${\displaystyle \mu ^{*}(E)=\inf \left\{\sum _{k}\ell (I_{k}):I_{k}{\text{ is a sequence of open intervals such that }}E\subseteq \bigcup _{k}I_{k}\right\}.}$

Units

In the physical sciences and engineering, when one speaks of

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

mile (mi). A unit of length used in navigation is the nautical mile (nmi).^[7]

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

fermi

(fm).

See also

• Arc length
• Humorous units of length
• Length measurement
• Metric system
• Metric units
• Orders of magnitude (length)
• Reciprocal length

References

Look up length, distance, width, or breadth in Wiktionary, the free dictionary.

Wikimedia Commons has media related to Length.

1. ^ "WordNet Search - 3.1". wordnetweb.princeton.edu. Archived from the original on 25 September 2016. Retrieved 15 March 2020.
2. ^ "Measurement: Length, width, height, depth | Think Math!". thinkmath.edc.org. Archived from the original on 24 February 2020. Retrieved 15 March 2020.
3. ^ History of Length Measurement, National Physical Laboratory Archived 2013-11-26 at the Wayback Machine
4. ^ Caldwell, Chris K. (1995). "Graph Theory Glossary".
5. ^ Cheung, Shun Yan. "Weighted graphs and path length".
6. ^ Le, Dung. "Lebesgue Measure" (PDF). Archived (PDF) from the original on 2010-11-30.
7. ISBN 9781852336820
   .