Degrees to Meters for Dummies, the Rewrite

Illustration of Great Circle Distances

More and more these days, the intern assigned to look for dead links is finding that Leaf Group has rolled out an "update" to one of the old eHow.com posts we'd already featured. Sometimes they're good enough to ignore, but way too often the rewrite specialist has done little more than reword the original, mistakes and all, and pad it a little. That was surely the case for Chris Deziel in his attempt to rework the Sciencing.com post, "How to Convert Distances from Degrees to Meters."

In the original, penned by some college boy pretending to be Emile Heskey,¹ the eHowian made the assumption that the OQ wanted to know how to calculate the travel distance between two (distant) cities for which you know the Latitude and Longitude. We rather doubt it; the OQ probably wanted to know how to find the length of an arc based on the angle or perhaps convert Lat-Long to UTM. It makes no difference, because "Heskey" totally hosed the process.

Deziel took what Heskey had written and ran with it. In addition to rewording the content slightly, he added some in tangentially-related mathematical mumbo-jumbo to calculate the length of a one-degree arc of any circle on the Earth's surface: "Converting NASA's measurement of the Earth's radius into meters and substituting it in the formula for arc length, we find that each degree the radius line of the Earth sweeps out corresponds to 111,139 meters." Which, we suppose, is correct: we didn't do the math. What Chris neglects to mention (perhaps because he didn't realize it) is that the number he cites is the length of 1⁄360th of the circumference of a great circle. That fact renders the remainder of his post suspect. Heck, it renders the remainder of his post Flat. Out. Wrong.  Here's what Deziel would have you do, given two sets of lat-long coordinates:

1. Determine the Separation of Latitude
2. Determine the Separation of Longitude
3. Convert the Degrees of Separation to Distances (multiply by 111,139)
4. Use the Pythagorean Theorem

Let's try an example, using the cities of Rome (41.90N, 12.49E) and Ottawa, Ontario, Canada (45.43N, 75.72W). That makes the latitude separation 3.53 degrees * 111139 = 392321 m; longitude separation = 88.21 degrees * 111139 = 9803571 m. The hypotenuse of a triangle with those sides is the square root of (392321² + 9803571²), or 9811598 meters (9812 km).

Unfortunately, Ottawa and Rome are actually 6737 km apart²... Why, you ask? Apparently, Chris forgot that lines of latitude and longitude do not form a rectangular grid; because a degree of longitude is only 111,139 meters long at the equator. To calculate the distance between any two points on the surface of the planet, you must determine how many degrees apart the two points lie on a great circle (see image). The only time you can perform that simplistic calculation is if the two points have the same longitude or if both lie on the equator. Anywhere else, you can't just close your eyes and pretend that latitude and longitude are a rectangular grid.
As Deziel himself said, "the Earth is basically a sphere." Unfortunately, our Dumbass of the Day seems to have forgotten that little complication.

¹ Apparently, the lad didn't think anyone at DMS was an English football fan
² If you don't believe us, use the measurement tool on Google Maps.

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