That is just a shortcoming of our mathematical working model. Just as simple geometry can describe area and volume of fixed geometrical shapes (circles, squared, rectangles, triangles), etc., it can only approximate that of more complex shapes.
It takes calculus to describe areas and volumes of irregular shapes through integration using limits. We just haven't discovered the type of mathematical model that will give us a finite solution for π.
Calculus was essential in the development of mathematical expressions for optical design, because you could dispense with actual geometrical "ray tracing" and simply apply 3rd order or higher equations using derivatives to obtain solutions without tedious logarithmic traces.
Our mathematical tools are incapable of solving for π; the exact solution of π does not prove or disprove π. It is a fact and it is provable without a single real number.
Regarding Scripture or Orthodoxy?
Observable, but not provable, at least not mathematically. It used to bother me that all our knowledge and I mean all human knowledge, is an approximation to truth and fact. We can observation and interpret and approximate with remarkable accuracy but every aspect of our live is lived with some gap of the undefined or unknown. This gap is closed with what theologians call faith and scientists call theory. It's entertaining to contemplate while fishing or smoking a nice cigar.