late 14c., rounden, "to make round, give roundness to," from round (adj.). Sense of "make a circuit round" is from 1590s. Sense of "bring to completeness" is from c. 1600; meaning "to approximate (a number)" is from 1934; with up or down, "to increase (or decrease) a number by adding to its last digit," by 1956. Meaning "turn round and face, turn on and assault" is from 1882. Related: Rounded; rounding.
Sense of "go past or get round" is by 1743. To round out "fill up" is by 1856. To round off is from 1680s as "make round, finish with a curved or rounded form;" by 1748 as "finish appropriately and neatly." Also compare roundup.