Curvature of the Earth

We all know that the surface of the Earth is curved, but exactly how ‘curvy’ is it?  At what scale do you actually need to start compensating for it in a construction project?  Let’s do some easy math and find out…

The Earth is not a sphere — it’s actually an oblate spheroid — but since oblate spheroid math requires a university degree to understand, we can simplify things by treating the Earth as a sphere.  That makes the calculations understandable by anyone who has passed, say, Grade 10 or 11 high school math.

The mean radius, r, of this simplified Earth sphere is 6,371,008m.

Let d be a straight-line distance that you project at a tangent from any point on the Earth’s surface:

Application of Pythagoras’ Theorem gives the following:

c² = a² + b²
(r+h)² = r² + d²
r+h = √(r² + d²)
h = √(r² + d²) – r

…where h is the height of the end point of that line above the surface of the Earth.

A distance of 1,000m is something that most people can relate to, and is probably near the upper limit of what you would use for large-scale projects, so I’ll use that distance to calculate the magnitude of curvature of the Earth.

Solving with d = 1,000m we get:

h = √(6371008² + 1000²) – 6371008
h = 6371008.078480516 – 6371008
h = 0.078480516m
h ≈ 78mm

So, to put this in a way that I hope everyone can understand:

The surface of the Earth is not now, nor has it ever been, flat —
it falls away from a perfectly straight line at the rate of 78mm every 1,000m.

Thanks to similar triangles and the metric system, you can interpolate easily from that number:

78mm @ 1,000m  ≡  7.8mm @ 100m  ≡  0.78mm @ 10m  ≡  0.078mm @ 1m

So, if you built a house that was 10m long on top of a perfectly flat concrete raft, and placed it on the surface of a perfectly spherical Earth, the curvature of the Earth would result in a 0.78mm gap under one edge of the raft.  It would take 8 sheets of 75gsm photocopy paper or 1 grain of coarse sand to fill that gap.

If a 0.78mm fall over 10m (or any of its equivalents) does not compromise the integrity of your construction, then you can safely ignore the curvature of the Earth — pretend that the Earth is flat* — and just build it.  No-one will suffer.

Now you know.   Happy building.  🙂

* Treating the Earth as flat for small-scale construction projects is perfectly fine.  However, believing that the Earth is flat means you’re joining the ranks of a bizarre religious cult — and that’s not a good idea.  The curvature of the Earth is hard to see with the human eye, and hard to measure with the sort of measuring devices ordinary folks can buy at a hardware store.  Just because it looks flat doesn’t mean it is.  The challenge folks have in measuring that curvature at small scales is due simply to a) the lack of precision and accuracy of their instruments, and/or b) their lack of mathematical skill.

To be fair, very few of us construct things on a scale where we need to factor in curvature of the Earth, so we can happily go through life ignoring its existence.  That doesn’t mean curvature doesn’t exist — it just means ordinary folk don’t need to calculate it, compensate for it, or even worry about it, in their day-to-day lives.  Rest assured, however, that every modern engineer that signs off on a bridge, stadium, dam, airport runway or terminal, canal, supertanker, cruise liner, cargo ship, hospital, aircraft carrier, freeway intersection, railway station, shopping mall, or tunnel has obtained super-accurate measurements and crunched the numbers to compensate for curvature of the Earth — to ensure that their constructions don’t fail and that people don’t die.

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2 Responses to Curvature of the Earth

  1. Miriam English says:

    I remember when I was a kid, reading in the May 1979 issue of Scientific American, that month’s Amateur Scientist column, titled “How to Measure the Size of the Earth with Only a Foot Rule or a Stopwatch”. I still have the issue.

    Here it is as HTML, for your pleasure:
    http://miriam-english.org/files/size-of-earth/How_to_Measure_the_Size_of_the_Earth.html

    I know it is only slightly related, but thought it might give you a nice buzz. 🙂

    • Tim says:

      I’m familiar with Eratosthenes’ work, but hadn’t come across the methods by Rawlins or Gerver before. It’s amazing what a ‘bit’ of skill in math will let a person do. Thanks for that!

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