To take off is to leave the ground and rise into the air. What a plane does at the end of the runway.
It splits round an object in a quite different sense, since taking off a coat means removing it, and that is the commoner everyday use. There are two more: something that takes off becomes suddenly successful, and taking somebody off means imitating them for comedy. The noun takeoff covers the aircraft sense and the impression. Time off work is annual leave rather than this phrase.