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**HyperKaehler** I am working through the book "Geometry - A High School Course" by Serge Lang, and the only problem I have gotten stuck on so far is in chapter 3, which concerns the Pythagorean theorem. The question is in two parts, the first of which I have no trouble with, but the second of which puzzles me:

*a. A ship travels 6 km due South, 5 km due East, and then 4 km due South. How far is it from its starting point? (The answer is NOT 15 km!) You may assume that the path of the ship lies in a plane.*

*b. If actually the ship starts at the North Pole, on earth, what would the actual distance be?*

For part a, I got the answer $\displaystyle 5\sqrt{5}$ km for the distance. But I am confused by the second part, which seems to add the curvature of the Earth into the equation. Earlier in the book there was a section on arcs which involved a couple of questions calculating latitude and longitude (and for which we were told to use 40,000 km for the circumference of the Earth). But there is nothing in it about relating distance on the Earth's surface in two dimensions to its central angles, which is the only thing I can think of doing.

I would really like to work this out for myself, but I keep getting confused with diagrams. Is it perhaps a simpler problem than I'm making it out to be? Thanks a lot!