Editing Length (section)

== Use in mathematics ==
=== Euclidean geometry ===
{{main|Euclidean geometry}}
In Euclidean geometry, length is measured along [[straight line]]s unless otherwise specified and refers to [[line segment|segments]] on them. [[Pythagorean theorem|Pythagoras's theorem]] 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 (triangle)|altitude]], a line segment drawn from a vertex [[perpendicular]] to the side not passing through the vertex (referred to as a [[Base (geometry)|base]] of the triangle), is called the height of the triangle.

The [[area]] of a [[rectangle]] is defined to be length&thinsp;×&thinsp;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&thinsp;×&thinsp;width.

The [[volume]] of a [[Rectangular cuboid|solid rectangular box]] (such as a [[plank of wood]]) is often described as length&thinsp;×&thinsp;height&thinsp;×&thinsp;depth.

The [[perimeter]] of a [[polygon]] is the sum of the lengths of its [[Edge (geometry)|sides]].

The [[circumference]] of a circular [[Disk (mathematics)|disk]] is the length of the [[Boundary (of a manifold)|boundary]] (a [[Circle (geometry)|circle]]) of that disk.

=== Other geometries ===
{{Further|Non-Euclidean geometry}}
In other geometries, length may be measured along possibly curved paths, called [[geodesic]]s. The [[Riemannian geometry]] used in [[general relativity]] 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 (graph theory)|cycle]], [[Path (graph theory)|path]], or [[Walk (graph theory)|walk]] is the number of [[Edge (graph theory)|edge]]s it uses.<ref>{{Cite web|url=https://primes.utm.edu/graph/glossary.html|title=Graph Theory Glossary|last=Caldwell|first=Chris K.|date=1995}}</ref> In a [[weighted graph]], it may instead be the sum of the weights of the edges that it uses.<ref>{{Cite web|url=http://www.mathcs.emory.edu/~cheung/Courses/323/Syllabus/Graph/dijkstra1.html|title=Weighted graphs and path length|last=Cheung|first=Shun Yan}}</ref>

Length is used to define the [[shortest path]], [[girth (graph theory)|girth]] (shortest cycle length), and [[longest path]] between two [[Vertex (graph theory)|vertices]] in a graph.

=== Measure theory ===
{{main|Lebesgue measure}}
In measure theory, length is most often generalized to general sets of <math>\mathbb{R}^n</math> via the [[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|open interval]] is first defined as
: <math>\ell(\{x\in\mathbb R\mid a<x<b\})=b-a.</math>
so that the Lebesgue outer measure <math>\mu^*(E)</math> of a general set <math>E</math> may then be defined as<ref>{{cite web|url=http://zeta.math.utsa.edu/~mqr328/class/real2/L-measure.pdf|title=Lebesgue Measure|last=Le|first=Dung|url-status=live|archive-url=https://web.archive.org/web/20101130171814/http://zeta.math.utsa.edu/~mqr328/class/real2/L-measure.pdf|archive-date=2010-11-30}}</ref>
: <math>\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\}.</math>