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So it's pretty well known that the area of a sphere is $4 \pi r ^2$..

I was trying to derive this using a projection from the surface of the great circle to the surface of the sphere and I end up getting $2 \pi r ^2$ which is of course wrong.

Here's the logic:

For each point $x, y$ on the surface of the great circle, map it to the point in 3D space that's the distance $r$ from the center of the sphere. Assuming the 3rd coordinate being $z$, There are two such points, one in each hemisphere: $(x, y, z)$ and $(x, y, -z)$.

Similarly, take any point $(x, y, z)$ on the surface of the sphere and map it to the point $(x, y, 0)$ on the surface of the sphere. This mapping is 1-1 for points on each hemisphere. Proof by contradiction: If you could map two points $(x, y, z_1)$ and $(x, y, z_2)$ from a single hemisphere (hence both $z_1 > 0$ and $z_2 > 0$ or the other way around), to $(x, y, 0)$ then by definition $z_1 = z_2$ otherwise these two points can't be the same distance $r$ from the center of the sphere.

So we've shown that the mapping is 1-1 in both directions, e.g. you can uniquely map each point on the surface of each hemisphere to one point on the surface of the great circle and similarly you can uniquely map each point on the surface of great circle to one point on the surface of each hemisphere.

With this established, the next part is clumsily something like below:

Given that the area of the great circle is $\pi r ^2$ and each point on it can be uniquely mapped to two points on the surface of the sphere, then it has to be that the area of the sphere is $2 \pi r ^2$.

Why is this logic wrong?

Paghillect
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    Welcome to Math SE. Consider the simpler $1$-dimensional case, in particular the $1$-$1$ mapping between $[0,1]$ and $[0,10]$ with $f(x)=10x$. However, the length of the two ranges are definitely different. – John Omielan Feb 19 '25 at 06:20
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    Once you have a line segment you have an uncountably infinite number of points -- the same number of points that there are in all of Euclidean space, and there is a one-to-one mapping between them (albeit not as neat and tidy as your mapping). – David K Feb 19 '25 at 06:24
  • @JohnOmielan It's a different enough question, since the other question appears to be a misunderstanding of a method that works (narrow circular ribbons interpreted as just circles) whereas this method is wrong in all kinds of ways. I'm pretty sure I've seen a duplicate, though. – David K Feb 19 '25 at 06:28
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    @DavidK Thank you for your feedback. I still think that Why can't we derive the formula for surface area of a sphere thus? is pretty similar, but on rereading what the OP wrote and the other post again, I think it's different enough that closing this as a duplicate of it may not be warranted, so I've retracted my close vote. Nonetheless, I agree there's likely a duplicate post so, if that's the case, I hope you or somebody else finds it. – John Omielan Feb 19 '25 at 06:33
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    I thought I might actually have answered a duplicate question some time ago. I found what appears to be the answer I was thinking of, but the question I was answering was yet another way to get a wrong answer (maybe the most confused yet). What I remembered was a comment elsewhere on the page about adding up the points, which wasn't in the question statement. I still have the nagging idea that someone sometime did a one-to-one mapping but I still haven't found it if it exists. – David K Feb 19 '25 at 06:48
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    @JohnOmielan I think the same as you; the other question was very similar, just not quite similar enough. In the meantime we may have answered in comments or via links to other questions (where the answer may serve even if the question isn't quite the same). – David K Feb 19 '25 at 06:54

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