Why does a sphere have the lowest surface area out of all 3D shapes for a given volume?
In: Mathematics
Think of it this way. If you start with a sphere and then poke any portion in or out, you create an irregularity which needs more material. The bit of material right at the corner or edge holds little volume and is wasteful. Eliminating all the corners and edges leaves no wasteful bits to remove.
Take a square and it’s perimeter for a given volume, not cut all the corners and now look at the perimeter for the octagon, and just keep cutting corners till you get a circle. It’s similar to how Pythagorean theorem show us a^2+b^2=c^2 c is always going to be shorter than A and B combined, think of the corners as A and B and the cut you make is C
The mathematical proofs are definitely not eli5. Here is a different way to think about it.
One could reduce the problem 2D : why does a circle have the least perimeter for the same area?
One could further reduce the problem to 1D: why is the shortest way between two points a straight line? Because every other way involves a detour from the shortest path.
Now generalizing to 2D: a circle is is the shortest way to “sweep” an area, because at every point we are the shortest distance from the center as possible. Every other way involves a detour i.e. the distance from center becomes larger than what is required to sweep the area. If this distance from the center becomes larger, then we are increasing the perimeter.
In the same way, we can generalize to a sphere in 3D.
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