Etymology
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height (n.)

Old English hiehþu, Anglian hehþo "highest part or point, summit; the heavens, heaven," from root of heah "high" (see high) + -itha, Germanic abstract noun suffix (as in width, depth; see -th (2)). Compare Old Norse hæð, Middle Dutch hoochte, Old High German hohida, Gothic hauhiþa "height." Meaning "distance from bottom to top" is from late 13c. Meaning "excellence, high degree of a quality" is late 14c. Century Dictionary says "there is no reason for the distinction of vowel between high and height. The modern pronunciation with -t emerged 13c. but wasn't established until 19c.; Milton used highth and heighth is still colloquial in English. Compare Dutch hoogte, Danish hjöde.

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Definitions of height

height (n.)
the vertical dimension of extension; distance from the base of something to the top;
Synonyms: tallness
height (n.)
the highest level or degree attainable; the highest stage of development;
at the height of her career
Synonyms: acme / elevation / peak / pinnacle / summit / superlative / meridian / tiptop / top
height (n.)
(of a standing person) the distance from head to foot;
Synonyms: stature
height (n.)
elevation especially above sea level or above the earth's surface;
Synonyms: altitude
From wordnet.princeton.edu