Measuring the Earth with a piece of string and a lot of maths.

By David Coulson, (c) 2016.

dtcoulson@gmail.com

There is a way of calculating the size of the Earth that doesn’t need expensive surveying equipment or impossibly long pieces of rope. All you need are a couple of friends to hold a piece of rope in front of you while you look out to sea from a high place. Oh, yes, that and a tonne of mathematics.

Stand at a high place where you get a good view of the sea that sweeps very far to the left and right. Hopefully there will be a sign telling you how high above the sea you are. If not, there are ways of estimating height.

Where I live, in Christchurch New Zealand, there is a cliff I can go to that gives me a view of the sea that extends at least 120 degrees from the seaward mountains to the north and another cliff to the south. If you run your eyes along the horizon from side to side, you get the impression that the horizon is dipping down at the sides, very slightly but very distinctly.

If you had a rope stretched in front of you across your view, you would be able to see the curvature clearly. And if you held a ruler up in front of you, you could measure the angular separation at the midpoint. You can use this number to calculate the diameter of the Earth. (You can use your thumb too. At arm’s length it covers about 1 egree or so, but this is approximate).

α

Time to put some maths in here. The triangle on the left gives us the angle theta and the length d in terms of r and h. You can then get the radius of the cone from the triangle on the left.

hr

rθCos

Tanrd

os radius Cone Cd

If you look down on the cone from above, you can see that the horizontal distance (at sea level) between the two lines at its widest point is dCos(ϴ) - dCos(ϴ)Cos(φ).

hr

rθCos

Tanrd

os radius Cone Cd

Alpha is the observed angle between the two lines.

11

Sind

CosCosd Tan

Sind

CosdTan

Alpha is the observed angle between the two lines.

θ

hr 22

rhr

r

Cos

rhr

rTan

rhr

rTan

22

1

22

1

11

Sind

CosCosd Tan

Sind

CosdTan

I’ve done this calculation for the cliff near my home. The height of the cliff I estimate to be 500 m, and the angle of view is about 120 degrees. Knowing that the radius of the Earth is about 6371km, I can calculate that the bulge angle is about three-quarters of a degree.

Cos

rhr

rTan

rhr

rTan

22

1

22

1

Now I want to ‘invert’ this procedure so that I can go from the observed angle back to the radius of the Earth. This is a monstrous formula so I want to introduce a new parameter, f, which is the ratio between the Earth radius and the observer’s height above sea level.

frhr

hf ,

Cos

rhr

rTan

rhr

rTan

22

1

22

1

φCosf

Tan f

Tan

φCos ff

Tan ff

Tan

φCos f

Tan f

Tan

φCos rfrr

r Tan

rfrr

r Tanα

2

1

2

1

2

1

2

1

11

1

11

1

11

2

1

2

1

2

1

2

1

22

1

22

1

f is very small, so f 2 is insignificant.

1

1 2

Cosf

2

1

2

1

Cos

Cosf

Since f is very small, the angle is also very small.

2

2

2 2

22

2

1

2

1

1

1

11

11

φCos

f Tan f Tan

φCosf Cot f Cot

φCosf

Tan f

Tanα

2

1

2

1

Cos

Cosf

This formula is approximate but is reliable over a good range of circumstances. The approximation requires the cliff height to be very small compared to the radius of the planet, which is quite natural. It will work well for all observation points within the Earth’s atmosphere. You could even use it from a plane in flight. The need to divide by the cosine of phi (half of the horizontal angle between reference points) means that the formula will break down when the angle approaches 90 degrees. However, some testing shows that the formula still works very well when the angle is in the high eighties. For most natural applications, the calculated radius will have as many significant figures to it as the angle you measured.

(To get the planet radius, just divide the cliff height by f )

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