Wrapping Around the World

Suppose a string is wrapped around the Earth at the equator. (Assume the Earth is a sphere with a smooth surface.) Now pretend that we pull the string outward from the earth’s surface so that the string is always one-inch away from the earth’s surface all the way around the equator. How much more string will be needed to complete this new circle?

(Source: Fostering Children’s Mathematical Power. An Investigative Approach to K-12 Mathematics, Arthur J. Baroody, with Ronald T. Coslick, c. 1998 by Lawrence Erlbaum Associates, Mahwah, NJ.)

Extension
Suppose you added one foot to the string that wrapped snugly around the world. If you made the fit snug except for a small gap, what could fit under the gap?