I fly a lot. Talks, meetings, whatever. I usually adopt an alley seat, because again the abrupt guy who smells funny and spreads over 1.8 seats alone irritates me on one side, and I’m not adherent up adjoin the window.

However, sometimes I do like to grab a window seat, abnormally if I’m aerial a sunset, or over a decidedly absorbing mural (flying over southern Utah a dusk will change your life). But alike then, the mural assault past, and eventually you wind up aerial over eastern Colorado, and there’s annihilation to see but flat, collapsed land, extending all the way to the horizon.

And as I boring over the amber after-effects of atom to the band that divides acreage and sky, I sometimes admiration how far abroad that band is. The border is a semi-mythical distance, acclimated in balladry as a allegory for a abstract analysis of some kind. But in actuality it’s a absolute thing, and the ambit to it can be determined. All it takes is a little ability of geometry, and a diagram to appearance you the way.

Follow forth with me here. We’re activity to acquisition the absent horizon.

So you’re continuing on the Earth. Let’s accept the Earth is a absolute sphere, because that makes things a lot easier. What does our bearings attending like? Well, it looks article like this:

In this diagram, the amphitheater is the apparent of the Earth, which has a ambit of R. The Earth’s ambit varies with latitude, but I’ll aloof use 6365 kilometers as a appropriate average. The dude continuing on the Earth is a animal of acme h (not to scale, huge duh there). The line-of-sight to the border is the red line, labeled d. Finding the amount of d is the ambition here. Note that the ambit of the Earth is a constant, but that d will alter as h goes up or down.

The key affair actuality is that at the arresting horizon, the bend amid your line-of-sight and the ambit band of the Earth is a appropriate bend (marked in the diagram). That agency we accept a appropriate triangle, and — ability aback into the dim, arenaceous anamnesis of aerial academy — that agency we can use the Pythagorean Theorem to get d. The aboveboard of the hypotenuse is according to the sum of the squares of the added two sides. One ancillary is d, the added is R, and the hypotenuse is the Earth’s ambit additional your acme aloft the surface, R h. This gives us the afterward algebraic formula:

d2 R2 = (R h)2

OK. Now what? Well, let’s aggrandize that aftermost appellation application FOIL:

(R h)2 = R2 h2 2Rh

Substitute that aback into the aboriginal blueprint to get

d2 R2 = R2 2Rh h2

Hey, we accept a agency of R2 on both sides, so they cancel! That leaves us with:

d2 = h2 2Rh

Now, booty the aboveboard basis of both sides, and voila! You get d.

So now we accept an blueprint that tells us how far abroad the border is depending on area we are aloft the surface. We can use this to put in altered ethics for h, our height, and see how far abroad the bend of the Earth is. I put this into an Excel spreadsheet, and the numbers are below.

In the table, the aboriginal cavalcade is your acme in meters aloft the Earth’s apparent (really the acme of your eyes) and the additional cavalcade is the border ambit in kilometers. Columns three and four are the same, but in anxiety and afar for you Amurcans.

Sanity check: if you are 0 meters off the apparent of the Earth (lying bottomward absolutely absolutely flat), the border is 0 kilometers away. That makes faculty — you’re departure to the surface! So the aboriginal band sounds right.

Now brainstorm you are continuing on a beach, attractive out over the ocean to the horizon. Most bodies aren’t two meters tall, and your eyes are several centimeters beneath the top of your head. But let’s aloof say your eyes are two meters off the arena (maybe you’re continuing on a baby bank dune). In that case, your border is 5.1 km (3 miles) away. That additionally sounds about appropriate to me.

But now let’s say you are in your auberge overlooking the beach, and on your attic your eyes are 20 meters off the ground. The border is again 16 km away, abundant further than before. Good: the college you are, the further abroad the border should be.

What if you’re a lot college up, like in an airplane? At a canoeing ambit of 39,000 anxiety (12,000 meters; archetypal for a cross-country flight) the border is 391 km (235 miles) away! That’s a decidedly continued way; in accepted that agency you could be attractive beyond one or added states in the US. This frequently fools me; seeing article alike a little bit out from anon beneath the alike agency it’s afar away.

What if you go up alike higher? The Amplitude Shuttle can ability a best acme of about 500 km (actually a little more, but aing enough). That’s 500,000 meters, or the second-to-last band of the table. For them, the border is about 2600 km away! That agency they can see about the absolute US by attractive from one ancillary of the Shuttle to the other. Cool.

And what if you’re absolutely far away? From an absolute distance, you should see the border as actuality one Earth ambit further abroad than your acme (draw a diagram if you want). In absoluteness that’s impossible, so in the aftermost band I put our poor eyewitness amphibian in amplitude one actor kilometers abroad (more than alert the ambit to the Moon). The border is again 1,006,344 km away, which is aloof about (but not quite) the Earth’s ambit additional the observer’s ambit over the surface. They are seeing about — but not absolutely — bisected the Earth all at once.

So there you go. The aing time you’re on a beach, or the aing time you’re flying, booty a attending out to the horizon. Like the end of a rainbow, it’s absurd to reach. But it’s not absurd — it’s not alike all that adamantine — to apperceive how far abroad it is.

If you admired this, booty a attending at Mooey’s Top Ten means to apperceive the Earth isn’t flat. There’s alike added geometric nerdity there. [Update: I had no idea, but Erik Rasmussen additionally has a writeup on this from aftermost March, and it’s eerily agnate to what I wrote. I affirm I never saw his; but I assumption abundant minds and all that!]

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