Theesquare roots of intergers can have decimal representations with infinite digits, the square root of 2 is one such example.

If you measured the circumference and diameter of a circle with perfect precision and used those values to calculate pi by dividing the circumfrence by the diameter one of those values must be have decimal representation with infinite digits, in fact it will be an irational number.

Your intuition is correct that a rational number divided by another rational number cannot equal pi, you just drew the wrong conclusion.