To create a circle from an edge loop in edit mode, there is the function under the specials menu. Access by W > Loop Tools > Circle.
But is there a Function to do the opposite?
By now, I pick a quarter of amount of the circle vertices, scale them to an axis by zero, and doing this for all four quarters of the circle. but this only works as long as the circle is aligned to the main axes and has a number of vertices that can be divided by four.