We know that for a cylinder $$ V = \pi r^2 h $$ This formula is easily visualized as a stack of $h$ circles with radius $r$.
However, as a little experiment with the goal of trying to think about things differently, I attempted to do the same, but using rectangles instead. Naturally, at least to me, I visualized a circular cylinder as rectangles, with width $r$ and height $h$, revolved around the center point of the cylinder.
I thought that one could simply then say that the volume should be the area of the rectangle * the circumference of the cylinder with this calculation. $$ V = rh \cdot 2\pi r = 2\pi r^2 h $$ Obviously, this seems to not be true and is what is getting to me. Can anyone explain why this is not true? What am I missing here?
To further clarify, let's simplify it so we can visualize it with something physical. Take your phone, or anything rectangular. Now rotate it 180 degrees. We just made a cylinder by rotating a rectangle, with width 2r and length h, (pi)r times. $$ V = 2rh * \pi r = 2\pi r^2 h$$