And now explain why a polygonal approximation is "more valid" than the proposed versionBecause the proposed version isnt measuring what it says it is measuring. Its a cognitive illusion. The perimeter (of the circle) was never being measured in the first place. The propsed version is designed to plant a false assumption. Though, the flaw with my statement was i should have said "to approximate the perimeter (of the circle), a possible way is to use the hypotenuse..." Basically, its just measuring the boxes perimiter that has been made to conform to the shape of a circle via a symmetrically path formed of infinite interwoven segments..
[update: added some clarification]
