Imagine I wrap a rope tightly around the Earth. Now suppose I make one small mistake: I should have placed the rope 1 foot above the surface.
How much extra rope would I need?
Surprisingly, I would need only 6.28 feet of extra rope.
What’s even more interesting is that if I did exactly the same thing around a ball, I would still need 6.28 feet of extra rope.
It doesn’t matter whether the object is the Earth or a small ball. Raising the rope by 1 foot always requires the same extra length: 2π feet, approximately 6.28 feet.
This is one of those simple mathematical results that is surprisingly difficult to guess intuitively.