Stand on the beach and the sea seems to go on forever. It doesn't — you can work out exactly where it stops, and the answer is much closer than it feels.
The curve here is drawn far rounder than real Earth, otherwise it would look flat and you would see nothing at all. The numbers, though, are the true ones for the height you picked.
At Earth's real size, your 4 km circle is about a fifth of one pixel wide on this screen — too small to draw. That is why it needs the magnifying glass.
| Where you stand | You see | Of the way round Earth | Of Earth's whole surface |
|---|---|---|---|
| On the beach (eyes 1.25 m up) | 4 km | 1 / 10,000 | 1 / 10,000,000 |
| On a 125 m hill | 40 km | 1 / 1,000 | 1 / 100,000 |
Climb from the beach to the hill and you see ten times further — but you gain a hundred times more ground, because the patch grows in two directions at once. Ten times taller in the distance you see, a hundred times bigger in the land you see.
One more thing worth knowing about the loop itself. That "all the way round Earth" is almost exactly 40,000 km — 40,000,000 metres — and that is not a lucky coincidence. When the metre was invented in 1793, it was defined so that ten million of them would fit from the North Pole to the equator — a quarter of the way around. So the loop measuring 40 million metres is the metre doing exactly what it was designed to do: it is a piece of the Earth, shrunk to hand size. (Later surveys found the original definition a whisker off, which is why the precise modern figure is 40,008 km — they corrected the unit, not the planet.)
And how did anyone measure the whole loop in the first place, two thousand years before satellites? How do we know how big the Earth is? — Rung 1 of the cosmic ladder →
Standing on the beach, the Earth feels endless. It isn't. You can see 4 km, and the whole planet is 40,000 km around — ten thousand of your horizons, laid end to end. That is a big number, but it is a countable one. A passenger plane flies your entire horizon in about half a minute, and could circle the whole planet in under two days.
This is the thing worth taking away from this page: the Earth is not infinitely big. It only feels that way because we never see more than a sliver of it at once. Every ocean, every forest, all the air you will ever breathe, and every person who has ever lived fits inside that one 40,000 km loop.
Which is why the astronauts who saw the whole thing at once all came home saying some version of the same sentence: it looked small, it looked fragile, and it was the only one out there. Nothing else on this entire site has a breathable sky or a drop of liquid water waiting for you. We only get this one. Take care of it.
If you happen to be near a sea shore, then it's easy: look at the sea horizon and you see about 4 km. Cross that distance 10,000 times and you would have gone all the way around Earth. There's not much between you and the sea horizon, so how far is it, exactly? Try the two-hills game.
Open any map with hills on it — paper or on a screen — and find two hills about 40 km apart. They need to be big enough to see each other over the curve at that distance — at least one of them 125 m above the surrounding ground, as you now know. Make it part of the next trip.
Stand anywhere with trees in sight and start counting them. Out loud, properly, no skipping. You will give up somewhere around forty, and everybody does.
Now the reason it is worth doing. Somebody counted the trees on the whole Earth — with satellites, not by hand — and got about three trillion. The Milky Way, our entire galaxy, has a few hundred billion stars in it. So there are roughly ten trees on Earth for every star in the galaxy.
You just gave up counting something there is more of than there are stars in the sky above you.
Your line of sight leaves your eye and grazes the Earth, making a right angle with the radius at the point it touches. That is a right-angled triangle, so Pythagoras does the rest:
Multiply out the long side and you get R² + 2Rh + h² on one side and R² + d² on the other. The R² cancels, and h² is so small next to 2Rh that dropping it changes nothing you could measure — a metre of height against 12,742 km of planet. What is left is the formula:
distance to horizon = √( 2 × R × h ) R = Earth's radius, 6,371 km h = your eye height
Both in the same units. Eyes 1.25 m up is 0.00125 km, so √(2 × 6371 × 0.00125) = 4.0 km. On a 125 m hill it is √(2 × 6371 × 0.125) = 40 km. Multiply the height by 100 and the distance only grows 10 times — square roots are stingy like that.
Real air bends light slightly downwards, which usually lets you see a few percent further than the formula says. On a hot day over cold water it can do stranger things and lift distant ships into view entirely.