Undergrad Flat surface to curved surface

[anilswipe]

Hello everybody,

Suppose I take a paper of say surface area A.

Then I would somehow (Do what it takes to do it; cut, fold whatever but no overlapping.) make an ideally and theoretically, biggest possible, perfect sphere out of it. Let's say the surface area of this sphere is A'.

Now how much is the difference between the surface area of the two?

Is there a general formula to find this?

In lay man's terms:
Suppose I take a plane paper and convert it into a sphere without overlapping, how much paper will be leftover? What is the generalized mathematical formula, if there is one, to find the difference between the surface areas of the two?

Thank you.

PS: Though I have chosen the suffix 'Intermediate', (assuming, possibly wrongly, that there may not be High school grade answers to this) I would gladly invite Basic High school grade answers if possible, to keep things simpler and I would invite higher grade answers, if absolutely necessary.

 

[PeroK]

You can't make a sphere from a flat piece of paper. For the same reason you can't flatten out a spherical surface.

That said, if you cut the paper up into small very pieces and stuck them all together, you could get an approximate sphere of the same total area.

 

[anilswipe]

That said, if you cut the paper up into small very pieces and stuck them all together, you could get an approximate sphere of the same total area.

I get it, but I suppose , that is not what I want to do and which also means that I have not been able to put my thought across. I apologize for that and would edit the post to that effect. Thanks for the reply.

 

[Gang]

OK. One more step into the topic.

Assuming the spere is perfect and made of 100 % of the original piece paper. It will have its radium, and its calculated surface area. On the other hand, the paper has its area before being turned into a sphere.

Will the calculated spheric area equal to the original area of the plane paper?

 

[jbriggs444]

OK. One more step into the topic.

Assuming the spere is perfect and made of 100 % of the original piece paper. It will have its radium, and its calculated surface area. On the other hand, the paper has its area before being turned into a sphere.

Will the calculated spheric area equal to the original area of the plane paper?

Are you trying to ask whether something like the Banach-Tarski paradox might apply?

In the Banach-Tarski paradox, one shreds a solid (3 dimensional) sphere into a finite collection of subsets and the re-assembles the subsets into two spheres, each equal in volume to the original.

Importantly, the subsets are sufficiently "weird" so that they are not Lebesgue-measurable. So one cannot argue that the measure of the union is necessarily equal to the sum of the measures.

Obviously we are talking about two dimensions here and Banach-Tarski works in three. And we have the niggling problem that we have flat sets that we are trying to plop onto a spherical surface. So the problem is not even well defined to start with. (How can you require an area-preserving flat to spherical mapping function without having a shape with a measurable area).