# Height

A cuboid demonstrating the dimensions length, width, and height.

Height is the measure of vertical distance, either how "tall" something or someone is, or how "high" the position is.

For example, "The height of that building is 50 m" or "The height of an airplane is about 10,000 m".

When the term is used to describe how high something like an airplane or a mountain peak is from sea level, height is more often called altitude.[1]

In a cartesian space, height is measured along the vertical axis (y) between a specific point and another that does not have the same y-value. If both points happen to have the same y-value, then their relative height equals to zero.

## Etymology

The English-language word high is derived from Old English hēah, ultimately from Proto-Germanic *xauxa-z, from a PIE base *keuk-. The derived noun height, also the obsolete forms heighth and highth, is from Old English híehþo, later héahþu, as it were from Proto-Germanic *xaux-iþa.

## In mathematics

In elementary models of space, height may indicate the third dimension, the other two being length and width. Height is normal to the plane formed by the length and width.

Height is also used as a name for some more abstract definitions. These include:

1. The altitude of a triangle, which is the length from a vertex of a triangle to the line formed by the opposite side;
2. A measurement in a circular segment of the distance from the midpoint of the arc of the circular segment to the midpoint of the line joining the endpoints of the arc (see diagram in circular segment);
3. In a rooted tree, the height of a vertex is the length of the longest downward path to a leaf from that vertex;
4. In algebraic number theory, a "height function" is a measurement related to the minimal polynomial of an algebraic number; among other uses in commutative algebra and representation theory;
5. In ring theory, the height of a prime ideal is the supremum of the lengths of all chains of prime ideals contained in it.

## In geosciences

### In geology

Although height is relative to a plane of reference, most measurements of height in the physical world are based upon a zero surface, known as sea level. Both altitude and elevation, two synonyms for height, are usually defined as the position of a point above the mean sea level. One can extend the sea-level surface under the continents: naively, one can imagine a lot of narrow canals through the continents. In practice, the sea level under a continent has to be computed from gravity measurements, and slightly different computational methods exist; see Geodesy, heights.

### In geodesy

Instead of using the sea level, geodesists often prefer to define height from the surface of a reference ellipsoid, see Geodetic system, vertical datum.

Defining the height of geographic landmarks becomes a question of reference. For example, the highest mountain by elevation in reference to sea level belongs to Mount Everest, located on the border of Nepal and Tibet, China; however the highest mountain by measurement of apex to base belongs to Mauna Kea in Hawaii, United States.

## In aviation

In aviation terminology, the terms height, altitude, and elevation are not synonyms. Usually, the altitude of an aircraft is measured from sea level, while its height is measured from ground level. Elevation is also measured from sea level, but is most often regarded as a property of the ground. Thus, elevation plus height can equal altitude, but the term altitude has several meanings in aviation.

## In human culture

Human height is one of the areas of study within anthropometry. While height variations within a population are largely genetic, height variations between populations are mostly environmental.[citation needed]

The United Nations uses height (among other statistics) to monitor changes in the nutrition of developing nations. In human populations, average height can distill down complex data about the group's birth, upbringing, social class, diet, and health care system.