Etymology
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terrace (n.)

1510s, "gallery, portico, balcony," later "flat, raised place for walking" (1570s), from French terrace (Modern French terasse), from Old French terrasse (12c.) "platform (built on or supported by a mound of earth)," from Vulgar Latin *terracea, fem. of *terraceus "earthen, earthy," from Latin terra "earth, land" (from PIE root *ters- "to dry").

As a natural formation in geology, attested from 1670s. In street names, originally in reference to a row of houses along the top of a slope, but lately applied arbitrarily as a fancy name for an ordinary road. As a verb from 1610s, "to form into a terrace." Related: Terraced.

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terrazzo (n.)
type of flooring material, 1893, from Italian terrazzo "terrace, balcony" (see terrace).
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Taiwan 
literally "platform bay" (perhaps with a sense of "port"), from Chinese tai "terrace, platform" + wan "bay." Related: Taiwanese.
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patio (n.)

1818, "an inner court open to the sky" in Spanish and Spanish-American countries, from Spanish patio probably from Old Provençal patu, pati "untilled land, communal pasture," from Latin pactum "agreement, contract, covenant," noun use of neuter past participle of pacisci "to covenant, to agree, make a treaty," from PIE root *pag- "to fasten." Another theory traces the Spanish word to Latin patere "to lie open." Meaning "paved and enclosed terrace beside a building" is recorded by 1941. Patio furniture is attested from 1924 in California newspaper advertisements; patio door by 1973.

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veranda (n.)

also verandah, 1711, Anglo-Indian, from Hindi varanda, which probably is from Portuguese varanda, originally "long balcony or terrace," of uncertain origin, possibly related to Spanish baranda "railing," and ultimately from Vulgar Latin *barra "barrier, bar." French véranda is borrowed from English.

That the word as used in England and in France was brought by the English from India need not be doubted. But either in the same sense, or in one closely analogous, it appears to have existed, quite independently, in Portuguese and Spanish; and the manner in which it occurs without explanation in the very earliest narrative of the adventure of the Portuguese in India ... seems almost to preclude the possibility of their having learned it in that country for the first time .... [Col. Henry Yule and A.C. Burnell, "Hobson-Jobson, A Glossary of Colloquial Anglo-Indian Words and Phrases," 1903]
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