Is there a way to precisely place the rotation point for an object?

I've wondered for a long if there is a way to place the point of rotation exactly where you want it without dragging it and eyeballing the position. Specifically, when I want to rotate duplicate objects around a circle as in the attached jpeg. As you can see, this example is "close enough" but not exact. I'd like to be able to mathematically place the rotation point exactly on the center point of my circle. Is there a command for this?