Guesstimating the radius of the Earth

 Photo taken in the Outer Hebrides by my brother-in-law

(The doctor one, not the Flat Earth one)

This photo was taken a few days ago by my brother-in-law Alex from the Outer Hebrides where he's off galavanting at the moment. You can clearly see that half the ship is missing indicating either

a) it's sinking
b) the Earth is round

Going with (b) for the time being we can actually come up with a pretty good guesstimate for the size of the Earth just from this picture.

Let's guess that the ship is 10km away. (That's not likely to be accurate but it's certainly more than 1km - the width of Lake Windermere - and less than 100km - the distance from Portsmouth to Le Havre). So $l=10000$. Now let's assume that the bottom 10m of the ship is missing from view (again, not likely to be accurate but good enough for an order of magnitude calculation). So $h=10$. Now chuck it into this equation which is easily worked out with a bit of trig
$$
r=\frac{l^2}{2h}
$$
and, hey presto, we get $r=5000000$ i.e the radius of the Earth is about 5000km (to within an order of magnitude). Nice!

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