The first large format wheel was designed and built by Ferris in Chicago in the United States about the time that the Eiffle tower was constructed in Paris, France in 1889. The purpose was to use a structure to allow its riders to enjoy views above the city offered by no other means since it was taller than the buildings around it. This concept had been done before, but not on such a large scale.
I assume that the circular shape of the the Ferris design had at least three main purposes.
Purpose 1. take visitors from the ground to the sky and return to the ground safely while giving them high views and excitement.
Purpose 2. it must look exciting, revolutionary, fun, dangerous, and amazing to capture the audiences attention.
Purpose 3. The circular shape makes for the most efficient structural design to suite the purpose.
If you take a look at a Ferris wheel, you will note similarities to a bicycle wheel with wire spokes. Focus on the spoke for a moment. How could something so skinny, not much bigger than a coat hanger wire, hold so much weight? The answer is tension, which translates into pulling rather than pushing.
The reason the circular shape is important is because the whole wheel is supported from its central axis. The tension members that reach from this center of the wheel are only in tension (being pulled) by the lower half of the wheel. The upper half of the wheel is an arch that is being supported by the lower half of the wheel. It is somewhat difficult to imagine, but if you think about it long enough it might make sense.
To help you visualize what is happening structurally, think about a working Ferris wheel and what would happen if you were to replace the tension members (all the spokes) with rope. If they only carry tension, and not compression, then the bottom tension member is carrying quite a bit of weight and the top one is not carrying anything at all.
The reason the circle make sense is because an arch can carry all its loads in compression along the outer rim of the circle. The top part of the circle is an arch. For an example, think of an archway made of bricks, an arch shaped bridge, or other arches that take advantage of this principal.
A simple rotating triangle might work, but would carry far fewer passengers because only the tips of the triangle would be able to reach a loading platform. A circle maximizes the number of passengers, which maximizes your clients and profits if you are running an amusement attraction.