Height is the measurement of vertical distance

Distance

Distance is a numerical description of how far apart objects are. In physics or everyday discussion, distance may refer to a physical length, or an estimation based on other criteria . In mathematics, a distance function or metric is a generalization of the concept of physical distance...

, but has two meanings in common use. It can either indicate how "tall" something is, or how "high up" it is. For example "The height of the building is 50 m" or "The height of the airplane is 10,000 m". When used to describe how high something like an airplane or mountain peak

Mountain peak

A mountain peak is the highest part of a mountain; a mountaintop; the summit.Mountain Peak may also refer to:* Blue Mountain Peak, Jamaica* Hole in the Mountain Peak, Nevada, USA...

 is from sea level

Sea level

Mean sea level is a measure of the average height of the ocean's surface ; used as a standard in reckoning land elevation...

, height is more often called altitude

Altitude

Altitude or height is defined based on the context in which it is used . As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context...

. Height is measured along the vertical (y) axis between a specified point and another point.
English high is derived from Old English hēah, ultimately from Proto-Germanic *xauxa-z, from a PIE

Pie

A pie is a baked dish which is usually made of a pastry dough casing that covers or completely contains a filling of various sweet or savoury ingredients....

 base *keuk-.