working backwards from $\pi r^2$

Question

I have been dipping my toes into a bit of calculus (through the better explained website), however I have become stuck on my understanding of the area of a circle. I understand that the formula for the area of a circle can be arrived at by visualising a series of infinitely thin circles which decrease in circumference, straightening them, and arranging them into a triangle with a base of $r$ and a height of $2\pi r$.

However, just to push my own understanding, I tried to work backwards from $\pi r^2$. This has been a nightmare to explain via text to my friends and family, so I will include a comic I put together which explains my thought process and also my confusion.

To be clear - I'm not looking to be provided with another method for arriving at the $\pi r^2$ formula (which is what everyone I've asked so far has done), I'm wondering what's wrong with the thought process which I intuitively engaged in, so I can try to correct any incorrect underlying assumptions/understandings which led to the incorrect intuition.

Answer 1

I think the area of your stacked semicircles is only $2r^2$ because of overlap of each semicircle at all points except the very center(vertical). With the most overlap being near the top and bottom.

To avoid this overlap you would have to straighten out the arcs and get the true $\pi r\times r$ area.

Answer 2

It is incorrect to calculate the area of the extended semicircle as $(\pi r)\times r$. This would be valid if you had a vertical line segment of length $\pi r$ and extended it a length $r$ horizontally, yielding a rectangle. For nonrectangles, multiplying lengths like that isn't valid.

For a simpler example, imaging you had a diagonal line segment of length $L$ and you extended it horizontally a distance $r$, resulting in a parallelogram. What would be the area of the resulting region? Not $Lr$.