Another formula for a cylinder's Volume??

Question

Ok, so I thought about this but it doesn't make any sense. Let's say we have a cylinder with height h and the radius of the base r. Let's say I cut the cylinder vertically so it traces a rectangular shape, which has length and width of h and 2r. If we rotate its area 180 degrees around itself it should make up the volume of the cylinder. So we have its Area = 2rh.

Now we multiply that with the Circumference of the base divided by 2, which represents the rotation by 180 degrees. We have:

$2rh * 2πr/2 = 2πr^2h \ne πr^2h$

It should have equaled $πr^2h$ but it's double that. Why does this happen?

Answer 1

There are several issues with this approach:

1. What does it mean to rotate that rectangle “around itself”? My guess (since you mention 180 degrees) is that you rotate it around one if its middle lines (more precisely, the middle line parallel to the sides with length $h$). It's a cylinder with the same height, yes, but the radius of the base is larger than the radius of the base of the original cylinder.
2. I see no reason to expect that you should get the volume of the cylinder multiplying its are by $\pi r$.

Answer 2

As Gerry points out, only the points on the outer edges of the rectangle are tracing circles of radius $r.$ The portion of the rectangle in the central axis of the cylinder doesn't move at all, for example.

In general, given any point of the rectangle, it will be a distance of $\rho$ from the cylinder's central axis for some $0\le\rho\le r,$ and will trace a half-circle of radius $\rho.$ Stretching out the half-circumference above each point of the rectangle, we have two mirror-image right-triangular wedges edge to edge. Each has a base of half the rectangle (with area $rh$) and height of $\pi r,$ so volume $\frac12\pi r^2 h,$ whence the total volume is $\pi r^2 h,$ as desired.