Length

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Length
Scale kilometres miles.svg
The metric length of one kilometre is equivalent to the imperial measurement of 0.62137  miles.
Common symbols
l
SI unit metre (m)
Other units
see unit of length
Extensive?yes
Dimension L

Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension 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.

Contents

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 Einstein's special relativity, length can no longer be thought of as being constant in all reference frames. 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 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, 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 volume of a solid rectangular box (such as a plank of wood) is often described as length×height×depth.

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

The circumference of a circular disk is the length of the boundary (a circle) 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 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. [4] In a weighted graph, it may instead be the sum of the weights of the edges that it uses. [5]

Length is used to define the shortest path, 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 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 is first defined as

so that the Lebesgue outer measure of a general set may then be defined as [6]

Units

In the physical sciences and engineering, when one speaks of units of length , the word length is synonymous with distance. 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 (symbol, m) and is now defined in terms of the speed of light (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 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 centimetre) and include the astronomical unit (au), the light-year, and the parsec (pc).

Units used to denote sub-atomic distances, as in nuclear physics, are much smaller than the centimetre. Examples include the dalton and the fermi.

See also

Related Research Articles

<span class="mw-page-title-main">Angle</span> Figure formed by two rays meeting at a common point

In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles are also formed by the intersection of two planes. These are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection.

Area Size of a two-dimensional surface

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 open surface or the boundary of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a curve or the volume of a solid.

Diameter Straight line segment that passes through the center of a circle

In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere.

<span class="mw-page-title-main">Integral</span> Operation in mathematical calculus

In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.

<span class="mw-page-title-main">Metric space</span> Mathematical space with a notion of distance

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.

<span class="mw-page-title-main">Volume</span> Quantity of three-dimensional space

Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units or by various imperial units. The definition of length (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid that the container could hold, rather than the amount of space the container itself displaces.

Height Measure of vertical distance

Height is measure of vertical distance, either vertical extent or vertical position . For example, "The height of that building is 50 m" or "The height of an airplane in-flight is about 10,000 m". For example, "Christopher Columbus is 5 foot 2 inches in vertical height."

Proper length or rest length is the length of an object in the object's rest frame.

Elementary mathematics

Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels.

<span class="mw-page-title-main">Area of a circle</span>

In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.

Stadiametric rangefinding

Stadiametric rangefinding, or the stadia method, is a technique of measuring distances with a telescopic instrument. The term stadia comes from a Greek unit of length Stadion which was the typical length of a sports stadium of the time. Stadiametric rangefinding is used for surveying and in the telescopic sights of firearms, artillery pieces, or tank guns, as well as some binoculars and other optics. It is still widely used in long-range military sniping, but in many professional applications it is being replaced with microwave, infrared, or laser rangefinding methods. Although much easier to use, electronic rangefinders can give away the shooter's position to a well-equipped adversary, and the need for accurate range estimation existed for much longer than electronic rangefinders small and rugged enough to be suitable for military use.

In mathematics, the Peano–Jordan measure is an extension of the notion of size to shapes more complicated than, for example, a triangle, disk, or parallelepiped.

Two-dimensional Euclidean space Geometric model of the planar projection of the physical universe

Two-dimensional Euclidean space or simply two-dimensional space is a geometric setting in which two values are required to determine the position of an element on the plane. The set of pairs of real numbers with appropriate structure often serves as the canonical example of a Euclidean plane, the two-dimensional Euclidean space; for a generalization of the concept, see dimension. Two-dimensional space can be seen as a projection of the physical universe onto a plane. Usually, it is thought of as a Euclidean space and the two dimensions are called length and width.

Lebesgue integration Method of integration

In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.

Pythagorean theorem Relation between sides of a right triangle

In mathematics, the Pythagorean theorem, or Pythagoras' 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 is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the legs a, b and the hypotenuse c, often called the Pythagorean equation:

In trigonometry, a skinny triangle is a triangle whose height is much greater than its base. The solution of such triangles can be greatly simplified by using the approximation that the sine of a small angle is equal to that angle in radians. The solution is particularly simple for skinny triangles that are also isosceles or right triangles: in these cases the need for trigonometric functions or tables can be entirely dispensed with.

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 interferometer methods based upon the speed of light.

Blaschke–Lebesgue theorem Plane geometry theorem on least area of all curves of given constant width

In plane geometry the Blaschke–Lebesgue theorem states that the Reuleaux triangle has the least area of all curves of given constant width. In the form that every curve of a given width has area at least as large as the Reuleaux triangle, it is also known as the Blaschke–Lebesgue inequality. It is named after Wilhelm Blaschke and Henri Lebesgue, who published it separately in the early 20th century.

A geometric separator is a line that partitions a collection of geometric shapes into two subsets, such that proportion of shapes in each subset is bounded, and the number of shapes that do not belong to any subset is small.

References

  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. Cardarelli, François (2003). Encyclopaedia of Scientific Units, Weights, and Measures: Their SI Equivalences and Origins . Springer. ISBN   9781852336820.