Answer
If the radius changes at $\frac{dr}{dt}$, then the area changes at the rate $2\pi r\frac{dr}{dt}$.
Work Step by Step
The area of a circle of radius $r$ is $A(r)=\pi r^2$. If the radius $r=r(t)$ changes with time, then the area of circle is a function $r$ and $r$ is a function of time $t$. So ultimately $A$ is a function of $t$. If the radius changes at $\frac{dr}{dt}$, then the area changes at the rate $2\pi r\frac{dr}{dt}$.
