How far away is the horizon?

You know, MA,

here's a pretty simple one that I just can't answer for myself. Just how far away is the horizon? Seems pretty simple, right? But all the science books give all the math about the height of the observer, square root and all that good stuff. From my calculations, the horizon is only 1.5 miles away. But visibility is sometimes said to be, for example, 10 miles on a clear day. So if I lie on the sand at PB and I see a boat that looks like it is sitting on the edge of the world, how far away is it? I really, really want to know.

-- Langston in the Valley

Of course we all know that if you're lying on the sand at PB, you can see only as far as the cooler of the person between you and the horizon. So that's three feet? Four feet? But if you insist... Had to wake up the geometry elves. They don't get much of a workout, so they're not usually sitting around with pencils sharpened, ready to be called into the game. Once they'd had their coffee, they took a vote on whether they should even bother helping someone who has so little respect for square roots.

If you're lying on the PB sand, let's say your eyes are maybe three inches above sea level. That's not really true, but let's say it is. The elves calculate your personal horizon at 0.67 (statute) miles away. Raise your head up to see over that cooler, so your eyes are about a foot above sea level, the horizon is now about 1.34 miles away. Stand on the cooler, six feet above sea level, 3.3 miles. You'd have to be about 60 feet above sea level to spot a duck 10 miles away.

We'd give you a handy formula to figure this out for yourself, Langston, but clearly you've got a thing about math. But for anybody else, the distance to the horizon is about 1.17 times the square root of your eye height (in feet) above sea level. Basically, what we're doing is calculating the length hypotenuse of a right triangle.

And I know already that I'll get letters from the pickers of nits. What we've given here is the geometric distance to a point on the horizon. Because of light refraction, we're able to see any object (e.g., a boat) that is slightly beyond the horizon. And the top of a ship's mast will come into view when it is farther away than the geometric horizon. There's a formula for that too, but the geometry elves are bored and have gone back to sleep.