Okay.
So we’ve got an entirely flat surface that also happens to be the exact same length as the earth’s surface.
If you had one continuous piece of string that went from one end of that flat surface to the other, and on one end there was attached a bell… would you be able to ring the bell by pulling on the other end of string?
Since we’re doing strings around the Earth, here’s the simplest, most unintuitive fact in geometry:
Say you have a string wrapped taut around the planet (purely spherical), like a belt. You want to raise that string up so that it’s one meter above ground all the way around the planet. How much more string do you need?
I’ll give you a hint. You don’t need to know the radius of the Earth to know the answer.
c = pi x dSo, to increase d by 2 meters (cause
d = 2 x r), that’s 2 X pi, or 6.28 meters for string?Correct.
To really emphasize it, the same amount of extra string would be needed if it was instead wrapped around a small marble at first and the diameter expanded by the same amount.
So I’m bad at math. Can you explain why we’ve decided to multiply pi by 2? Is there an articulable reason or is it just a rule?
c+x= pi * (d+2) in this case, right? So where did that multiply pi by 2 come from?
Distribute the pi on the right side of your equality, and replace c with pi*d:
c+x = pi*(d+2)
pid+x = pid + pi*2
x = pi*2
To generalize for an height h,
x = pi2h
Edit: I did some weird markup, but won’t be fixing it
Oh, I see now. I missed some pretty basic math there with that distribution. That makes sense now, thank you!