14 January 2014

Calculating latitude in your head

Yesterday I was visiting with an old friend who now lives in Hong Kong. “What is Hong Kong’s latitude,” I asked, vaguely aware it was near the Tropic of Cancer. She said she couldn’t remember, but that Hong Kong to Bali was a four-hour flight, and Bali, due south of HK, was on the equator.

That was enough. Assuming the airliner averages 500 mph, the distance is 2,000 miles. One degree of latitude is 60 nautical miles. Our math put HK 33 degrees north of the equator. Reckoning that was too far north, I estimated 25–30 degrees to correct for the difference between nautical and statute miles.

Back at home, I checked the internet for the precise figures:

So our estimate of 25–30 degrees north for HK was approximately 12 percent off. That’s not unreasonable for a numbers crushed in the mind calculation. But it also was a lucky calculation because we assumed Bali was at the equator when actually it’s just over eight degrees south of the equator. Hong Kong and Bali are separated by just under 31 degrees of latitude.

So, I tried the calculation using different assumptions (and a calculator). A Boeing 767 cruises at 530 mph, or 470 knots. Four hours at 470 knots is 1,880 nautical miles. Dividing 1,880 by 60 nm per degree returns 31.3 degrees. Subtracting Bali’s latitude from 31.3 degrees returns 22.9 degrees for Hong Kong, an error of just 2.6 percent.

An error that small is gratifying, but misleading. My assumptions were reasonable, but flight times and airspeeds can vary, so a 10 percent margin of error is closer to what to expect.

If you try this, remember that a nautical mile is 1.15 statute miles, so that a degree of latitude averages approximately 69 statute miles.