Writer's Reference: Distance to the Horizon

[You've entered the Giveaway in Support of YA Asian Book Covers, right? Today might be your last chance! (Boy, I should've thought of a better name).]

How far is it to the horizon? How far away can you see an approaching object? This is something I come across in Air Pirates a lot, but it always takes multiple clicks and conversions to get at the simple formula I want. So, fully expecting mathematics to drive away half my audience, here it is (in both kilometers and miles):

So someone 5 and a half feet tall (1.7 meters) would see the horizon disappear about 3.1 miles (4.7 km) away.

Keep in mind:
  1. These are approximations. Don't be doing science with these numbers.
  2. These numbers only apply in clear weather.
  3. You could still see tall objects peeking over the horizon. More info on that below.

Measuring the distance to something over the horizon only requires one extra variable. When the top of the object first peeks over the horizon, you can figure out how far away it is like this:

Now if you wanted to figure out how far away an object is based on how much of it is peeking over the horizon . . . well, you're on your own. I love math and all, but I'd lose the other half of my audience if I did any more of it in public.

Boy, I hope at least one of you cares about this.


Matthew MacNish said...

I'd imagine standing on the deck of an airship makes one quite a bit taller.

Susan Kaye Quinn said...

Well, you know I care! Sometimes I wonder if it's a handicap that I care too much. I'm a stickler for getting the physics right (unless we're tossing out the handbook), while others are having fun making up stuff that sounds cool, but defies gravity (and not on purpose).

Steve MC said...

I don't know the math, but when reading up on sailing, I found a chart and copied out the numbers for how far you can see at what heights.

What's cool is how you can see just as far on the other side of the curve as you can on this side.

And this should be a t-shirt, or a cool poster in schools:

Don't be doing science with these numbers.

Sarah Ahiers said...

oh my god. This is so nerdy. I mean, awesome. But nerdy.
Seriously though, that is some frickin awesome research

Authoress said...

Ooooo, I have to tell you...I researched something similar for my dystopian. Basically, I needed to know at what point the rising sun would peek through a crack at a certain height at the top of a wall. I finally realized I was getting WAY too technical about my timing and gave it up. o_O

MattyDub said...

Well, to find out how far away something is based on how much of it is visible over the horizon, you'd have to know the height of that object. So it's probably less useful in general. But this was a great post! It's the sort of thing you have to think about when you're trying to create a plausible world. Other examples include: "How fast can Hobbits walk in a day?" and "How far can the family of a dwarf push him before he just can't take it anymore?" Although the math on those is a little harder. :)

Angela Brown said...

Hmm...I hope it's okay if the mathematics sort of, um, flew over my head. However, I did wax poetically in my head of the following: "The distance to the horizon only matters if you're NOT willing to take the journey." I have no idea if someone more famous than me said something similar to this before, but that's what came to me. And by more famous, I actually mean famous since I've yet to reach any level of fame :-)

Sarah said...

I'm impressed, Adam. This is nerdishly awesome. And good to know, should I ever need to calculate the distance to the horizon on a clear day. Thank you. And: "Doing math in public" is my new favorite phrase.

Daisy Carter said...


I feel very much like Peppermint Patty. "The answer is false! Oh, this is math? The answer is twelve!"

A.L. Sonnichsen said...

So much to think about. Yikes. I think this is why I write about "real life." Ha ha ha! My brain is not as big as yours, Adam.

Deniz Bevan said...

Love stuff like this :-)

Deniz Bevan said...

Just a question - if you've got a couple with an almost-one-foot height difference, would a certain destination appear further to one or the other? I guess it's all elementary, since the taller guy is going to be able to walk there more quickly anyway :-)