In sphere $r \propto \frac{1}{A}$! How is this possible? What's the wrong here? Surface area $A$ and volume $V$ of a sphere of radius $r$ are
\begin{eqnarray} A=4\pi r^2,\\
V=\frac{4}{3} \pi r^3.
\end{eqnarray}
But then
\begin{align} \frac{V}{A} & = \frac{r}{3}\\                                          
\Rightarrow r &= 3 (\frac{V}{A})\\                                
\Rightarrow r &\propto \frac{V}{A}
\end{align}
so, $r \propto  \frac{1}{A}$.
But how is it possible? I can realize $r \propto  A$. If $r$ increases $A$ will also increase. But what is the wrong in this calculation. Please help me.
 A: Last step is not correct, since $V$ depends on $r$.
A: If you hold $V$ fixed, then the radius $r$ of the sphere varies inversely with the surface area $A$.
A: You need to understand more accurately what "proportional to" means.  If you say that $f\propto g$, this means that $f$ and $g$ are quantities depending on one variable, say $x$, and there is a fixed number $c$ such that $f(x)=cg(x)$ for all values of $x$.
In your statement that $r\propto\frac{1}{A}$, presumably the variable should be $r$.  The statement is not justified by the equation $r=\frac{V}{A}$, because if $r$ is a variable then $V$ is also a variable, not a fixed number.
Your statement that "$r\propto A$ [because] if r increases A will also increase" is also wrong - proportionality is not just a matter of "increasing together".  In fact, $A=\pi r^2$, and $\pi$ is a fixed number, so you could say
$$A\propto r^2\ ,$$
or you could write the equation as
$$r=\frac{1}{\sqrt\pi}\sqrt A\ ,$$
so
$$r\propto\sqrt A\ .$$
But it is not true that $r$ is a fixed number times $A$.
