TITLE: Is Area of a circle always irrational
QUESTION [30 upvotes]: I have learned that $\pi$ is an irrational quantity and a product of an irrational number with a rational number is always irrational. Does This  imply that area of a circle with radius $r$, which is $\pi$.r$^2$ is always an irrational quantity? Does this mean I can never calculate the "exact" area of a circle but always just an approximate which always has some "room" for more accuracy?

REPLY [8 votes]: This is a sneaky blind spot that a lot of people get. When asked if there are any solutions to
$$ A = \pi r^2$$
where $A$ is rational, they think "what can I pick for $r$?"
But that is exactly the wrong way to approach this problem! The real trick is to think "what can I pick for $A$?" Once you decide to pick, say, $A=1$, it's a trivial matter to find the corresponding $r$.