Contents

1. Overview
2. Etymology
3. Cultural Impact

Oh, this again. Fine. If you insist on quantifying the void, let’s at least do it with some semblance of precision. You want to understand the measure of distance? Very well. Try to keep up.

Measure of distance in physical space

This article, for those who struggle with nuance, concerns a fundamental physical measurement . For other, less precise interpretations, you may consult Length (disambiguation) .

“Width” redirects here, because apparently, we need distinct terms for every slight variation. For other uses, see Width (disambiguation) .
“Breadth” also redirects here. For the specific, rather quaint measurements applied to watercraft, refer to Breadth (nautical) .

Length

The metric length of one kilometre — a distance some humans still find impressive — is functionally equivalent to the rather more convoluted imperial measurement of 0.62137 miles . One could argue which system is more prone to existential despair, but that’s a debate for another time.

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

Length, at its core, is a quantitative expression of distance . It’s how we articulate how far one point is from another, a concept seemingly simple, yet it underpins nearly all physical understanding. Within the rigorous framework of the International System of Quantities , length is formally recognized as a quantity endowed with the singular dimension of distance. Most systems of measurement — those attempts to bring order to chaos — invariably select a base unit for length, from which the myriad other units are then, often awkwardly, derived. In the globally accepted International System of Units (SI), the foundational base unit for length has been designated as the metre . Perhaps you’ve heard of it.

It is, for the average observer, commonly understood that length denotes the most extended dimension of a static object. However, such simplistic definitions rarely survive contact with reality. This understanding is, in fact, not universally applicable and can be surprisingly dependent on the orientation or relative position of the object in question. What you perceive as “long” might become “tall” with a simple rotation, demonstrating the rather arbitrary nature of human perspective.

Various specific terms are employed to describe the length of a fixed object, each carrying its own contextual baggage. These include height , which is explicitly defined as vertical length or vertical extent, typically measured from a designated base. Then there’s width and breadth, terms often used interchangeably, generally referring to a dimension shorter than the primary length. And finally, depth, which quantifies the measure of a third dimension , often implying penetration or distance inward from a surface. It’s a rather elaborate vocabulary for what fundamentally describes an extent in space.

To reiterate, with the patience I rarely possess, length quantifies a single spatial dimension. This contrasts sharply with area , which inconveniently measures two dimensions (expressed as length squared, naturally), and volume , which demands a measure of three dimensions (length cubed, for those keeping score). The progression is rather straightforward, if you bother to pay attention.

History

The very act of measurement became crucially important the moment humans decided to abandon their nomadic wanderings and embrace the dubious comforts of settled lifestyles. This shift necessitated the use of building materials, the allocation of land, and, inevitably, the initiation of trade with neighboring groups. As interactions and commerce between disparate communities burgeoned, so too did the pressing need for standardized units of length . Without them, chaos, or at least rampant disagreement, would ensue. Later, as society inconveniently evolved into a more technologically driven entity, the demand for significantly higher accuracy in measurement proliferated across an increasingly diverse array of fields. This ranges from the microscopic intricacies of micro-electronics to the truly vast, almost comically distant, scales required for interplanetary ranging. Humanity, it seems, just loves to measure things.

Then, Einstein arrived, and everything became delightfully more complicated. Under his theory of special relativity , the very notion of length could no longer be considered constant across all reference frames . The implication? A ruler that precisely measures one metre in one frame of reference will, quite inconveniently, not be one metre long when observed from a reference frame that is in relative motion to the first. This means the perceived length of an object is not an absolute truth, but rather a variable dependent on the speed of the observer. A rather elegant way to tell us that our perceptions are inherently flawed, wouldn’t you agree?

Use in mathematics

Mathematics, bless its heart, attempts to bring structure to these concepts.

Euclidean geometry

Main article: Euclidean geometry

In the rather comfortable and predictable world of Euclidean geometry , length is, unless explicitly stated otherwise, measured along straight lines and refers specifically to segments of those lines. Pythagoras’s theorem , that rather famous relationship concerning the length of the sides of a right triangle , stands as but one of the countless applications of length within Euclidean geometry . However, length is not exclusively confined to the rectilinear; it may also be measured along various other types of curves, a concept then referred to as arclength . It’s a distinction that often escapes the casual observer.

Within the confines of a triangle , the length of an altitude — that particular line segment meticulously drawn from a vertex perpendicular to the opposing side (which is rather uncreatively called a base of the triangle) — is known as the triangle’s height. A surprisingly straightforward application of the concept, for a change.

The area of a rectangle is, as any geometry student should know, precisely defined as the length multiplied by the width of that rectangle. If, purely for illustrative purposes, one were to orient a particularly long, thin rectangle upright on its shorter side, its area could then be equally, if somewhat redundantly, described as its height multiplied by its width. The underlying quantity remains the same, only the labels shift.

The volume of a solid rectangular box – envision a robust plank of wood for practicality – is typically described by the product of its length, height, and depth. A simple enough calculation, yet crucial for anyone attempting to build anything more complex than a sandcastle.

The perimeter of a polygon is nothing more than the sum of the lengths of its individual sides . It’s merely a journey around its boundary, if you must think of it in such prosaic terms.

And the circumference of a circular disk ? That’s simply the length of its boundary , which, as you might infer, is a perfect circle . Basic, yet often overlooked.

Other geometries

Further information: Non-Euclidean geometry

Beyond the comforting straight lines of Euclid, in other, often more complex geometries , length may be measured along paths that are, by their very nature, curved. These specific paths are termed geodesics . Riemannian geometry , which forms the mathematical backbone of general relativity , serves as a prime example of such a geometry, where the shortest path between two points isn’t necessarily a straight line in the traditional sense, but a geodesic following the curvature of space-time itself. In the more tangible realm of spherical geometry , length is ascertained along the great circles on the surface of a sphere. The distance between any two points on this sphere is then defined as the shorter of the two possible lengths along the great circle that is uniquely determined by the plane passing through those two points and the very center of the sphere. It’s all rather elegant, if you appreciate such things.

Graph theory

In an unweighted graph , the length of a cycle , a path , or even a simple walk is determined by the mere count of the edges it traverses. A straightforward tally, no complex calculations required. However, in the more sophisticated domain of a weighted graph , the length transforms into the sum of the numerical ‘weights’ assigned to each of the edges it utilizes. A rather crucial distinction, if you’re attempting to, say, optimize a network.

This concept of length is then employed to rigorously define several fundamental properties within a graph: the shortest path between two vertices , the girth (which designates the length of the shortest cycle), and, conversely, the longest path connecting any two vertices within the graph. These are not trivial distinctions; they are the very scaffolding of network analysis.

Measure theory

Main article: Lebesgue measure

Within the realm of measure theory , the concept of length is often generalized to encompass more complex, arbitrary sets within the multidimensional space of $$\mathbb{R}^n$$ through the application of the Lebesgue measure . In the specific, less complicated one-dimensional case, the Lebesgue outer measure of a given set is defined with respect to the lengths of what are called open intervals . To be excruciatingly concrete, the length of an open interval is initially defined with a straightforward subtraction:

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

From this foundational definition, the Lebesgue outer measure, denoted as $$\mu^*(E)$$, for a more general set $$E$$ can then be precisely formulated as:

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

If that doesn’t make immediate sense, perhaps you’re simply not trying hard enough. It’s a rather elegant way to assign a “size” to sets that defy simple geometric description.

Computer science

Main article: String (computer science)

In the less abstract, more practical domain of computer science , the length of a string is defined with admirable simplicity: it is merely the total number of characters that constitute that string. For example, that rather uninspired bit string you see below:

$$\texttt{10010100001}$$

This particular sequence, thrilling as it is, possesses a length of precisely 11. It’s not rocket science.

Many programming languages , in their endless quest for convenience, incorporate a specific string data type, often equipped with a built-in method (typically named length or size) designed to effortlessly retrieve the length of any given string.

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String s = "Hello, world!";
int len = s.length(); // len will be assigned the value 13. Shocking, I know.

Furthermore, the “length” of a sequential data structure — such as a dynamic array or a list — is simply described as the total number of individual elements it currently contains. Again, not exactly a profound mystery.

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import java.util.List; // Because apparently, we need to be explicit about everything.

int[] a = {1, 2, 3, 4, 5};
int len = a.length; // len will be 5. The number of elements.

List<Integer> list = List.of(1, 2, 3, 4, 5, 6, 7, 8);
int len = list.size(); // len will be 8. No surprises here.

Units

Main article: Unit of length

In the rather serious fields of the physical sciences and engineering , when one refers to units of length , the word “length” is, for all practical purposes, synonymous with distance . There exists a bewildering array of units employed to measure length, a testament to humanity’s historical inability to agree on anything. Historically, these units of length often originated from rather arbitrary sources: the lengths of human body parts (hardly consistent, one might argue), the distance covered in a certain number of paces, the span between prominent landmarks or geographical points on Earth, or, perhaps most amusingly, simply the length of some common, readily available object. Precision was clearly not a priority.

In the universally preferred and significantly more rational International System of Units (SI), the foundational base unit of length is the metre (represented by the symbol ’m’). This unit is now precisely defined not by some royal foot, but in terms of the immutable speed of light in a vacuum (approximately 300 million metres per second ). Derived from the metre , the millimetre (mm), centimetre (cm), and the kilometre (km) are also commonly used units, offering convenient scales for various measurements.

In stark contrast, the U.S. customary units , often referred to as the English or imperial system of units , still cling to their own set of commonly used units of length: the inch (in), the foot (ft), the yard (yd), and the mile (mi). A rather stubborn adherence to tradition, wouldn’t you agree? For specialized applications like navigation , the nautical mile (nmi) is employed, because why make things simple?

For those who enjoy trivial conversions, remember that 1.609344 kilometres are equivalent to 1 mile . You’re welcome.

When contemplating the truly staggering distances encountered in the vastness of space, as is common in the field of astronomy , the standard metre or kilometre simply don’t suffice. Here, units are scaled up dramatically, encompassing the astronomical unit (au), the light-year (a rather poetic unit, if you think about it), and the parsec (pc). These units, while less relatable to our terrestrial experiences, are essential for making any sense of the cosmic scale.

Conversely, for the minuscule, almost incomprehensible distances encountered at the sub-atomic level, such as within nuclear physics , units are scaled down far beyond the utility of the millimetre . Examples include the fermi (fm), a unit so small it makes the nanometre look positively gargantuan. It seems humans are equally obsessed with the infinitely large and the infinitely small, and inventing new units to describe them.