In a greek theater, the round flat space where the action took place was called the

It's called The Orchestra.

Explanation:

The orchestra was the central circular space where the choir, formed by dancers, performed. The thymele was a stone set in the center of the orchestra intended for the offerings to the god Dionysus. The proscenium was for the choir leader, the choir leader, the space between the stage and the orchestra, and the stage, initially made of wood and later of stone, was the space for the exhibition of costumes and the exchange of them and masks.

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