There are different ways to define pi. In real analysis, you first define the exponential function: exp(x) = sum (x^k)/(k!), then cosine: cos(x) = Re(exp(i*x)), where i is the imaginary unit, and then show that cos(x) has exactly one root in [0,2], which you call pi/2.
Your statement suggests that the definition via the circle is more fundamental that other definitions, which it isn't, e.g. because it requires a very special metric in Euclidean space (of which there are infinitely many), while real analysis only requires a metric on the real numbers.
Your statement suggests that the definition via the circle is more fundamental that other definitions, which it isn't, e.g. because it requires a very special metric in Euclidean space (of which there are infinitely many), while real analysis only requires a metric on the real numbers.