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New Math Formula
3.1416 is rational.
In fact, I dare you to write an irrational number without using a symbol. (Well, outside of the character set of 1-9 and maybe you can use a decimal point too).
From you descriptions of this equation it looks like you are being distracted by the tautology.
r^2 * PI = (r * sqrt(PI))^2
is valid for any precision of PI, not because PI is poorly defined, but rather its true for PI because it is true for all non negative numbers.
What you are forgetting is that the area of a circle is ONLY r^2 * PI when PI is its irrational value. We get 'close enough' by approximating PI.
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