Testwiki:Reference desk/Archives/Mathematics/2022 June 12

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June 12

Accuracy of flat-earth approximation

I know that, for short distances, we can get away with pretending that the Earth is flat. For example, if I measure 3 meters in a horizontal line, then turn 90 degrees and measure 4 meters in another horizontal line, then the distance to my starting point will be for all intents and purposes 5 meters. What about longer distances? If I were to measure 3000 meters horizontally, and then turn 90 degrees and measure 4000 meters horizontally, then to what level of (im)precision would I need to be working to get away with saying that the distance from my starting point to my ending point is 5000 meters? If I am working, say, to the nearest centimeter, then at what distance can I no longer get away with pretending that the Earth is flat? 32.219.123.165 (talk) 05:49, 12 June 2022 (UTC)

You can use the Spherical law of cosines for this problem, or more simply the spherical analogue of Pythagoras' theorem, $\cos c/R=\cos a/R\cos b/R$, where R is the radius of the earth and a, b and c the (great circle) lengths of the sides, with c as the "hypotenuse". Applying this to a 3-4-5 triangle I calculate that for sides of 300km and 400km the error in c is about 0.02%, and for 3000km and 400km it's 2.49%. To get an error of 1% we need sides of about 1950 and 2600. AndrewWTaylor (talk) 06:44, 12 June 2022 (UTC)

The following table assumes a perfectly spherical Earth with a radius of 4000 miles. The side lengths, measured on the surface along a great circle, are in miles:

      a     b     c
      3     4     4.9999997
     30    40    49.9997
    300   400   499.7
   3000  4000  4657
   6000  8000  6401
   9000 12000  3599

--Lambiam 08:05, 12 June 2022 (UTC)

Fun fact:

       a            b            c
   75398.22369 100530.96492      0.000000000000

 --Lambiam 08:38, 12 June 2022 (UTC)

For a concrete answer to the specific question about working to the nearest centimetre, I take the Earth radius to be Template:Val, the largest of the radii commonly used in mathematical models. The hypotenuse of a flat right triangle with sides Template:Val and Template:Val equals Template:Val. But the hypotenuse when laid out on our orb is only Template:Val, which, rounded to the nearest centimetre, is Template:Val. An error of one 23995th part of one degree in the right angle gives an equally large discrepancy.  --Lambiam 15:00, 13 June 2022 (UTC)