First of all, amphitheaters and circuses were designed for purely visual spectacles while theaters were the structures designed with acoustics in mind. Vitruvius book 5 has a decent description of how theaters were designed (but it should be noted that no known Roman theater was built exactly according to Vituvius's prescriptions.) The basic element was circular seating on a slope that helped sound waves from the stage travel farther without dispersing. I don't have the scientific knowledge to go into anymore detail on how the shape helps sound waves travel. There's also evidence that limestone used in the construction of some theaters helped by dampening lower frequency background noise. The Greeks built their theaters by carving them into a hillside. The theater of Dionysus in Athens was carved into a slope at the base of the acropolis. The Romans either built temporary theaters with wooden bleachers or permanent theaters made of concrete, but they would often carve into a hillside to support the lower levels of seating if an appropriate hillside site could be found. They also used bronze vessels known as echea that would supposedly help resonate the sound. Whether these actually helped is unknown.

First of all, amphitheaters and circuses were designed for purely visual spectacles while theaters were the structures designed with acoustics in mind. Vitruvius book 5 has a decent description of how theaters were designed (but it should be noted that no known Roman theater was built exactly according to Vituvius's prescriptions.) The basic element was circular seating on a slope that helped sound waves from the stage travel farther without dispersing. I don't have the scientific knowledge to go into anymore detail on how the shape helps sound waves travel. There's also evidence that limestone used in the construction of some theaters helped by dampening lower frequency background noise. The Greeks built their theaters by carving them into a hillside. The theater of Dionysus in Athens was carved into a slope at the base of the acropolis. The Romans either built temporary theaters with wooden bleachers or permanent theaters made of concrete, but they would often carve into a hillside to support the lower levels of seating if an appropriate hillside site could be found.