How is the circumference of the earth calculated?

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Answer 1 · 2016-11-16T02:01:49

From visible horizon.

From visible horizon:

Let h be the height of the flag mast of a ship, from sea level. If d is

the maximum sea-level distance from which the top could be seen.

the circumference of the Earth is nearly

$2pi(d^2/h+h)$ .

Explication:

If h = 20 meters = 0.02 km and d = 11288 meters,

the radius of the Earth

$= d^2/h+h=$

$= (11.288)^2/0.02+0.02$

$=6370.967$ km

The circumference

= $2pi$ x (radius)

$=40029.966$ km, nearly

Proof:

d=(radius)(angle subtended at the center, in radian)= $Ralpha$

Here the proportion $alpha/R$ is very small.

If T is the Top of flag mast, O is the observer, C is the center of the

Earth,

$cos alpha = 1-alpha^2/2=1-(d/R)^2$ , nearly

$= (CO) / (CT)= R/(R+h)=(1+h/R)^(-1)=1-h/R+(h/R)^2$ , nearly,

as h/R is very small.

So, the approximation formula is

$d^2/R^2=h(R-h)/R^2 to R = d^2/h+h$ , nearly.