Puzzle - Wrap a Rope Around the Earth

Image from: http://what-if.xkcd.com/imgs/a/67/expanding_rope.png

This is a simple puzzle that forces us to think about proportion.  There are lots of versions of it out there, but the first appearance was as early as 1702 in Williams Whiston's "The Elements of Euclid" (here's a link to the 1714 edition).  

-Imagine a ball that you can hold in your hands.  Let's say for this puzzle that it is a globe of the world.  You have a length of rope that fits around the globe, completing the full length of the circumference.  Now let's say that you want another piece of rope to wrap around the globe, but at all times that piece of rope lies 1 meter away from the surface of the globe (so it wraps around the globe, but there is 1 meter of space in between the rope and the globe at all points).  How much longer will this rope need to be than the first one?

-Got that?  Well now, let's take the puzzle a little further:

-Imagine a piece of rope that wraps around the entire Earth at the equator.  At about 40,075 km that's a pretty long piece of rope, but it's not big enough!  Now let's say that you want to cut another piece of rope so that, just like in the first part of the puzzle, this new piece of rope wraps around the Earth at the equator but with 1 meter of space between the rope and the Earth's surface.  How much longer must this new rope be when compared to the rope that was wrapped around the globe with 1 meter of space from the first part of this puzzle?







You may need to refresh your circle geometry to find the answer.  Also, this picture and the link in the caption might help if you're stuck:

An answer to a similar phrasing of the puzzle (as well as this image) can be found at http://puzzles.nigelcoldwell.co.uk/fortyone.htm 

(If the link above didn't help, here's another that will show you how to find the answer)