Is Area of a circle always irrational?

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Cordell Crosby · Accepted Answer

Just because   A  =  π      r    2   has a   π, which is irrational, does not mean that   A has to be irrational. This is because       r    2   could be irrational. For example, take   a  ,  b  ∈      Z   to be any positive integers with   b  ≠  0. We can form the fraction   a      /    b. Then taking   r  =                    a                  b          π                    , we have  A  =  π      r    2    =  π                              a                      b            π                                2    =                    a        π                    b        π              =            a      b      which is certainly rational.Essentially, this results from the fact that product of two irrational numbers need not be irrational. We take the irrational number π and multiply by the irrational number       r    2   and get a rational. Another example would be 2–√⋅2–√=2. However, it is certain that if       2    ⋅      2    =  2 were rational then       r    2   would be rational and then the area   A would be irrational.

Is Area of a circle always irrational?

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