>>13619085I don't know what "arbitrarily small" means.
Strips have length and width and shape.
If you are asking whether the sum of the differential lengths (where the width of the strips goes to zero) of strips cut from a square and a circle of equal area are equal, the answer is yes.
If you did it in practice, there comes a point where the strip is thin enough that any error would be unmeasurable or lost in the noise of the deformation of the material.
Calculus works with a choice, not a law. You choose to accept that the error goes to zero even though you could also choose that it never made it to zero. That is what a limit is: a choice.
However in practice, the error gets lost in the measure, and the choice of letting the limit be equal to zero pays off for most applications involving matter, space, and time.
>>13619778That area doesn’t conserve perimeter is irrelevant if you are cutting the shapes up into strips because you are changing the perimeter with every cut strip. As the strips become thinner they approach a differential with is all perimeter and no area, which is the limit. The sum of those all-length-no-width theoretical strips would be equal regardless of the shapes you started with. Theory becomes practice when the error is so small that you can't measure it anymore.
So the shapes you start with are irrelevant. Only the area matters. If the areas are equal, the sum of the lengths of the strips approach the same length as you make the strips thinner and thinner, regardless of the shapes you start with.
