volume cylinder

Question

It's a silly question, but still:

Is it possible to calculate the volume of a cylinder only knowing the area $A_1$? (I don't know what it's called. It equals $r \cdot h$.)

Answer 1

No.

The volume of a cylinder is $V = \pi r^2h = \pi(rh) r = \pi A_1 r$.

Even if you fixed the value of $A_1 = rh$, you could vary $r$ ($h$ would change in inverse proportion to keep $A_1$ constant) and that would cause $V$ to change. The point is that, without knowing the value of $r$ (and $h$) separately, you cannot pin down the value of $V$.

Answer 2

No, the area is proportional to $rh$ and the volume to $r^2h$. You can't solve for two parameters with a single equation.

Answer 3

Yes, by the theorem of Pappus, we can compute a volume of revolution by multiplying the area being swept by the length of the path traveled by the center of mass of the area.

We have the center of mass of the area is at $(\frac{r}{2}, \frac{h}{2})$ and travels a length of $2\pi\frac{r}{2}=\pi r$. Since the area of $A$ is $rh$ we get $rh(\pi r)=\pi r^2h$