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Based on the tunnel distance formula from Wikipedia, I calculate that the tunnel distance (shortest distance between two points on Earth's surface, straight through Earth, based on a spherical Earth) between the South Pole and a point 40 degrees north is about 90% that of the full diameter of Earth. I personally find that surprising. I believe the formula on Wikipedia is correct, both because I did a few sanity checks as well as found this answer with the same formula.

My question is, how can it be that a point which is so far still from the North Pole (it is closer to the equator than it is the North Pole) be almost as far away from the South Pole as the North Pole is? I believe my calculation, but the result is a bit surprising, and I have a superior who doesn't quite believe the result. How would I explain it to him? He is mathematically literate.

When I say 40 degrees north, I am talking about standard latitude and longitude, which is measured from the equator.

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    $\begingroup$ what do you mean by "a point 40° north?" 40° north with respect to the south pole is still in the southern hemisphere. That distance cannot be 90% of the diameter. With respect to the south pole, this point will have to be in the northern hemisphere $\endgroup$ – imranfat Sep 3 '15 at 21:00
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    $\begingroup$ Draw a circle, with South Pole and the point at $40^\circ$ North indicated. From the point of view of distance from the South Pole, the circle kind of flattens out as you go further North. (As you get not too far from the North Pole, the distance is increasing quite slowly.) $\endgroup$ – André Nicolas Sep 3 '15 at 21:14
  • $\begingroup$ @imranfat, I mean standard latitude and longitude, which is understood to be measured north and south from the equator. $\endgroup$ – NeutronStar Sep 3 '15 at 21:26
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The angle at the South pole between the line to the North pole and the line to a point at $40^\circ N$ is $25^\circ$. D is a point at $40^\circ$N. BC is the diameter of the earth, which we can take as $1$. BD is your tunnel, of length $\cos 25^\circ \approx 0.906$ Your calculation is correct.

enter image description here

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It's not actually that surprising. Just draw an approximate picture: circle with lines

You can see the two lines are actually pretty close in length.

Note that this doesn't mean the path from the North Pole to 40 degrees is 9 times shorter than the path from the South Pole. It's due to the fact that both paths go along a non-optimal straight-line path, so the sum of their distances will be a lot more than the length of the diameter.

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