If $R \sim \text{Uniform}(0, 1)$, and I define a circle to have radius $R$, and I define a RV $A$ as the area of the circle, it follows that $A = \pi R^2$.
It took me a while to grasp why $A \sim \text{Uniform}(0, \pi)$ is not true, intuitively. I think I understand that $A$'s pdf $f_A(a)$ is not systematically mapping $a$ to a probability, because the probability of a specific point in a uniform distribution is $0$.
Instead, I turned to computations to solve for $f_A$.
$f_A(a) = f_R(r)$ where $r = \sqrt{\frac{a}{\pi}}$. But this get's to a similar point where I think that $A$ would be distributed uniformly. Could someone clarify why this is incorrect and how you generally should approach finding the pdf of a function of a random variable with a known disribution.