Math CentralQuandaries & Queries


Question from Renee, a student:

Assuming that the earth is a sphere of radius 6378 kilometers, what is the difference in latitudes of two cities, one of which is 600 kilometers due north of the other?

What a strange thing to assume :-P

Oh, it's true (close enough for rock'n'roll); but it's not the fact that is easy to remember, historically interesting, and makes the math easy for this problem. That fact is:

The kilometer is by its original definition 1/40000 of the circumference of the Earth through the poles, or one ten-thousandth of the distance from the equator to the north pole. It's still very close to that, just defined in ways that are more precise and easier to measure.

So one degree of latitude (measured N-S) is $111 \frac19$ kilometers; a kilometer, in a N-S direction, is 0.009 degrees. So now you can solve your question easily.

A degree of longitude, measured E-W, is shorter except near the equator. A kilometer from the North Pole there are 360 degrees in a 1-km-radius circle, so each one is only about 160m long! Up around 45N where I live, a degree of longitude is about 78 km. Very close to the equator a degree of longitude is actually longer because the earth's spin makes it slightly M&M-shaped.)

Good Hunting!

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