North Pole, 40 degrees north, and South Pole: question on why first two seem so close relative to third?

Question

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.

Answer 1

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.

Answer 2

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

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.