Definitions

# Length

[lengkth, length, lenth]

Length is the long dimension of any object. The length of a thing is the distance between its ends, its linear extent as measured from end to end. This may be distinguished from height, which is vertical extent, and width or breadth, which are the distance from side to side, measuring across the object at right angles to the length. In the physical sciences and engineering, the word "length" is typically used synonymously with "distance", with symbol $l$ or $L$.

Length is a measure of one dimension, whereas area is a measure of two dimensions (length squared) and volume is a measure of three dimensions (length cubed). In most systems of measurement, length is a fundamental quantity, from which other quantities are defined.

## Units of length

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. Units of length may be based on 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 fixed object.

In the International System of Units (SI), the basic unit of length is the metre and is now defined in terms of the speed of light. The centimetre and the kilometre, 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, the foot, the yard, and the mile.

Units used to denote distances in the vastness of space, as in astronomy, are much longer than those typically used on Earth and include the astronomical unit, the light-year, and the parsec.

Units used to denote microscopically small distances, as in chemistry, include the micron and the ångström.

## Length of moving rods

While the length of a resting rod can be measured by direct comparison with a measuring rod, this comparison cannot be performed while the rod is moving, due to relativistic concerns. In this case we define its moving length as the distance between its two endpoints at a given instance.

If the world lines of the two endpoints of the rod expressed in the coordinates of an $R ,$ inertial reference frame are

$mathbf x_1\left(t\right) = \left(t,x_1\left(t\right),y_1\left(t\right),z_1\left(t\right)\right),$

and

$mathbf x_2\left(t\right) = \left(t,x_2\left(t\right),y_2\left(t\right),z_2\left(t\right)\right),,$

then the length of the rod in this reference frame at the $t ,$ instance is

$l_R\left(t\right) = sqrt\left\{ left\left(x_2\left(t\right)-x_1\left(t\right)right\right)^2 + left\left(y_2\left(t\right)-y_1\left(t\right)right\right)^2 + left\left(z_2\left(t\right)-z_1\left(t\right)right\right)^2 \right\}.$

Since in special relativity the relation of simultaneity depends on the chosen frame of reference, the length of moving rods also depends.

## Generalisations

In the differential geometry of curves and differential geometry of surfaces, length is an important global Riemannian invariant.