is there any size of a circle where the area is an integer and the radius is an integer?

Question

The formula for the area of a circle is pi times the radius squared. The radius is the diameter divided by 2. Imagine a line, like the axis, but instead, it doesn’t go past the edges. Now, the length of that line is the diameter.

Answer 1

This is not possible. To prove it, we will proceed by contradiction.

Let $A=n=\pi R^2$ and let $R=m$ where $n,m$ are positive integers. Then $n=\pi m^2$, and so $\pi = \frac{n}{m^2}$. Therefore $\pi$ is rational, and we have a contradiction. Therefore, both of the claims (integer radius and area) cannot simultaneously hold.

(I.e. note that it is of course possible to have simply the radius an integer, or instead the area integer-valued, but this would necessarily make the other irrational.)