From Greek 'theatron' (θέατρον, a place for viewing), from 'thea' (a seeing), from PIE *dʰeh₁- (to see). Originally meant just the seating area, not the stage. 'Theory' is its cousin — both from the same Greek root for seeing.
A building or outdoor area in which plays and other dramatic performances are given; the activity or profession of acting in, producing, directing, or writing plays; a room for lectures with seats in tiers.
From Latin 'theatrum', from Greek 'theatron' (θέατρον), meaning 'a place for viewing,' from 'theasthai' (θεᾶσθαι, 'to behold, to watch'), from 'thea' (θέα, 'a view, a seeing'), ultimately from PIE *dʰeh₁- ('to see, to look'). The word originally referred to the seating area of a Greek amphitheater — the part where the audience sat and watched — not the stage. The Greeks distinguished the 'theatron' (viewing area) from the 'skēnē' (stage building, source of English
The Greek theater gave English three words from three parts of the same building: 'theater' (from theatron, the seating area where you watch), 'scene' (from skēnē, originally a tent or hut behind the stage where actors changed masks), and 'orchestra' (from orchēstra, the circular floor where the chorus danced, from orcheisthai 'to dance'). Most remarkably, 'theory' is a cousin of 'theater' — Greek 'theōria' meant 'a looking at, contemplation,' from the same root 'thea' (seeing). A theory is, etymologically, a way