Proofing for the surface area of a sphere

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Hello, I tried to logically think of the surface area of a sphere and came up with πr2πr or 2π^2r^2. I know that this is wrong and that the formula for SA is 4πr^2 but in my mind the formula that I came up with (because I needed it for a calc test but I forgot) should work. My reasoning for this:

Take half the circumference of a circle to form an arc (πr). Then, from the top middle point of the arc, drag the arc out of the page in a circular motion and back into the page to form a sphere. The distance traversed at the point you moved the arc should be the circumference (2πr) in theory right? Shouldn't multiplying these two values give the surface area of a sphere?

I kept on thinking about this today and it bothered me because I couldn't figure out what the flaw was with the way I was thinking this through. It would be great if someone could point out was is fundamentally wrong with the way I derived this formula.

 

[ksdhart2]

There's two ways you can think about the surface area of a sphere. The first is a method first used by Archimedes, who discovered that the surface area of a sphere is the same as the lateral surface area of a cylinder having the same radius as the sphere and a height the length of the diameter of the sphere. You can see an explanation of his work here. The other way uses calculus, and examines the relationship between volume and surface area - namely that the surface area of a sphere is the derivative of the volume. This page on Stack Exchange has several posts explaining why that is. I found one post particularly interesting that notes that this relationship also holds for a square and a cube, although that may not be readily apparent:

The circle (and sphere) is not really that special. It also works for the square if you measure it using not the side length s, but half that, h=s/2. Then its area is A=(2h)² = 4h² with derivative \(\displaystyle \dfrac{dA}{dh} = 8h\), which is its perimeter.

This paper offers a more in-depth examination of the solids for which the area is the derivative of its volume.