Length

• 1. Introduction

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  Length is one of the oldest things humans ever bothered to measure. It is the measure of distance, the answer to how far one point sits from another in physical space. In the International System of Quantities, length is a quantity carrying the dimension of distance. Most systems of measurement begin by choosing a single base unit for length, then derive every other unit from it. In the International System of Units, that base unit is the metre.

  Most people picture length as the longest dimension of a fixed object, its most extended reach. The source warns that this is not always true. The longest dimension can shift depending on the position the object is held in. So a quantity that feels simple turns out to have edges. What happens to length when the object lies down instead of standing up? What happens when the observer is moving? And how does one idea stretch from the size of an atom to the gulf between stars?

• 2. Words For A Dimension

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  Height is vertical length, the vertical extent of a thing. The source ties it to a base from which vertical measurements can be taken. Without that base, the word loses its footing.

  Width and breadth usually point to a shorter dimension than length. They describe the across, not the along. Depth gets its own job, naming the measure of a third dimension, the reach into a thing rather than across or up it.

  Length itself measures only one spatial dimension. Area measures two, which is why it is length squared. Volume measures three, which is why it is length cubed. The same root word climbs from a single line to a surface to a solid, gaining a power at each step. That climb sets up the harder question of how length behaves once it leaves a single straight line.

• 3. Geometry And Curved Paths

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  In Euclidean geometry, length is measured along straight lines unless stated otherwise, and refers to segments on those lines. Pythagoras's theorem, which relates the lengths of the sides of a right triangle, is one of many places length does real work in this geometry. Length can also run along curves, where it is called arclength.

  An altitude of a triangle is a line segment drawn from a vertex, perpendicular to the side that does not pass through that vertex. That side is called a base, and the altitude's length is called the height of the triangle. So height, the word from vertical extent, reappears here as a precise geometric quantity.

  The area of a rectangle is defined as length times width. Stand a long thin rectangle on its short side and the same area can be read as height times width. The volume of a solid rectangular box, such as a plank of wood, is often described as length times height times depth. The perimeter of a polygon is the sum of the lengths of its sides, and the circumference of a circular disk is the length of its boundary circle.

  Other geometries let length follow curved paths called geodesics. The Riemannian geometry used in general relativity is one such example. In spherical geometry, length runs along the great circles of a sphere. The distance between two points on the sphere is the shorter of the two great-circle lengths, fixed by the plane through both points and the center of the sphere.

• 4. Counting And Strings

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  In graph theory, length stops being a smooth distance and becomes a count. In an unweighted graph, the length of a cycle, path, or walk is simply the number of edges it uses. In a weighted graph, length may instead be the sum of the weights of those edges.

  Shortest path, girth, and longest path all lean on this idea. Girth is the shortest cycle length, and the longest path runs between two vertices in the graph. Length here becomes a tool for navigating networks rather than measuring rulers.

  Measure theory generalises length to broad sets through the Lebesgue measure. In the one-dimensional case, the Lebesgue outer measure of a set is built from the lengths of open intervals. The length of a single open interval is defined first, and the outer measure of a general set is then defined from those pieces.

  In computer science, the length of a string is the number of characters it contains. A bit string can have length 11. The string Hello, world! has length 13. Many programming languages give their string data type a method to retrieve this length. The same notion stretches to a sequential data structure such as a dynamic array, where length is the number of elements it holds, so the array 1, 2, 3, 4, 5 has length 5.

• 5. From Atoms To Astronomy

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  In the physical sciences and engineering, the word length is synonymous with distance, and several units exist to measure it. Historically, units of length came 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 from the length of some common object.

  The metre, symbol m, is the SI base unit of length, now defined in terms of the speed of light, which is about 300 million metres per second. The millimetre, centimetre, and kilometre are derived from it and widely used. In U.S. customary, English, or imperial units, the common units are the inch, the foot, the yard, and the mile. Navigation adds the nautical mile.

  For the vastness of space, astronomers reach far past the metre and kilometre. They use the astronomical unit, the light-year, and the parsec. For sub-atomic distances in nuclear physics, the units shrink far below the millimetre, down to examples such as the fermi.

  Measurement mattered the moment humans settled from nomadic life and began using building materials, occupying land, and trading with neighbours. As trade between places grew, so did the need for standard units, and a technological society now demands far higher accuracy across fields from micro-electronics to interplanetary ranging. Yet Einstein's special relativity unsettles the whole picture. Length can no longer be treated as constant across all reference frames. A ruler one metre long in one frame will not be one metre long in a frame moving relative to it, so an object's length depends on the speed of the observer.