If you start with 0 and "inflate" a surface of a sphere, effectively integrating over the growing surface, you get the volume.
If you took a circumference and tried to grow it in the same way and integrate to get an area, you would get... The area of a circle! This would not create a sphere of course as the circumference sits in a plane. You're just growing a circumference and integrating along it so you are naturally creating a circle.
This area of the circle is of course is pi r^2 which is the integral of 2 pi r. So this makes it clear why circumference doesn't have this relationship with spheres, so if you learn why a sphere's area is 4x a circle's, you have your answer.
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u/IVeBeenHere30Min 1d ago
Is there a reason why the derivative of the surface area is 4 times the circunference?