Building designed for performances, or part of one (12)
I believe the answer is:
amphitheatre
'building designed for performances or part of one' is the definition.
(cryptic definition)
This is all the clue.
(Other definitions for amphitheatre that I've seen before include "Roman tiered arena" , "Arena; gallery tier" , "Large oval stadium with tiers of seats" , "Classical home for drama" , "Theatre gallery seating area" .)