I'm trying to prove, without the aid of the isoperimetric inequality, that for any compact manifold with given surface area $S$ in $\mathbb{R}^3$, that the volume $V$ must be bounded. By this I mean that any sequence of homeomorphisms that also preserve $S$, can't result in an unbounded volume $V$.
In particular, I'm searching for an elementary proof of a weaker-than-isoperimetric inequality:
$$ V \leq f(S) \tag{*}$$
I tried to find such an inequality using the Pappus Centroid Theorem but I have not been successful so far.
Note 1: I am interested in a result that generalises easily to compact manifolds in $\mathbb{R}^n$ that aren't necessarily differentiable.
Note 2: My motivation comes from a problem in mathematical physics where an isoperimetric inequality is deduced so I don't want to assume the isoperimetric inequality beforehand.