Surface area of sphere coming out as $\pi^2 r^2$

Question

Take a hemisphere and divide its surface area into strips like on a watermelon. Each strip can be approximated as a triangle with the long two sides = $\pi \frac r2$ (quarter of circumference) and if they are laid out and rotated $180$ degrees alternatively to form a rectangle with sides $\pi\frac r2$ and $\pi r$ like in the derivation of the circle's area. Then the area of the full sphere comes out to $\pi^2r^2$. what is wrong here?

Answer 1

Comment only.

Not clear. You wanted to extend to 3d from the following 2d situation? May be Pappu's thm would help if location of CG is known.

1

Answer 2

If you could lay down on a plane your strip with lateral sides of length ${\pi\over2}r$ and base $b$, then you would find it isn't a triangle, but rather a figure as shown below. At a distance $h$ from the base its width is $b\cos(h/r)$, hence a simple integral gives its area as $rb$, instead of ${\pi\over4}rb$.

This ratio of ${\pi\over4}$ between the two areas is exactly the ratio between your result $\pi^2r^2$ (for the area of the sphere) and the right one.