# I'm sick of the city.

May 4, 2008 6:11 PM Subscribe

How far would I need to travel from a tower 100 feet high before the sight of it would become obscured by the horizon?

I'm wondering, also, where in America I could go and have literally no break in my line-of-sight. I'm tired of being crowded in by buildings.

I'm wondering, also, where in America I could go and have literally no break in my line-of-sight. I'm tired of being crowded in by buildings.

The

posted by johnvaljohn at 6:19 PM on May 4, 2008

**very**first result googling for horizon+formula:. It gives the formula and even answers your question directly.posted by johnvaljohn at 6:19 PM on May 4, 2008

**krisak**, I'm 28 years old. Just wondering, and my Googlefu failed me (horizon + formula! Face meet palm).

posted by grrlaction at 6:25 PM on May 4, 2008

Ah, I'm a cynic and pessimist. :) Hence my immediately assuming the worst.

I assumed that you were talking about land only. I would think the great lakes are the correct answer, but if you wanted land, South Dakota is flat and doesn't have many trees.

Florida, while having the lowest highest point, may or may not be flatter - it's certainly more populated, both with trees and people.

posted by krisak at 6:32 PM on May 4, 2008

I assumed that you were talking about land only. I would think the great lakes are the correct answer, but if you wanted land, South Dakota is flat and doesn't have many trees.

Florida, while having the lowest highest point, may or may not be flatter - it's certainly more populated, both with trees and people.

posted by krisak at 6:32 PM on May 4, 2008

There are stretches on I-70 on your way across Kansas, where the road is so flat, and extends pretty much straight for as far as you can see, that it's very difficult to judge how far away you are from, say, the Rocky Mountains, which is the only thing in your line of sight that you can be sure is farther away from you than the road you're on.

However, by the time you do reach the mountains, your fascination with this phenomenon will most likely have worn off.

posted by bingo at 6:46 PM on May 4, 2008

However, by the time you do reach the mountains, your fascination with this phenomenon will most likely have worn off.

posted by bingo at 6:46 PM on May 4, 2008

The Bonneville Salt Flats are pretty flat. Not much to do out there but race, though.

posted by SPrintF at 6:52 PM on May 4, 2008

posted by SPrintF at 6:52 PM on May 4, 2008

I just spent 20 minutes figuring out why my derivation was wrong, so I'm going to explain the whole thing now that I am thoroughly annoyed with myself (spoiler: radians, not degrees):

The Earth can be thought of as a circle with radius 6378.1 km, or about 20 925 525 feet. If you imagine your tower being at the North Pole, the point you are looking for is where a line drawn from the top of the tower intersects the circle at only one point - the tangent line. If you draw a line from the center of the Earth to this tangent point and another from the center of the Earth to the base of the tower, you have a right triangle; one property of a tangent line is that it is perpendicular to the radius of the circle that goes through the point where it intersects the circle.

Now you need to consider which distance you're looking for - line-of-sight distance from your eye to the top of the tower, or the distance you would need to walk to reach the tower. They're about the same for short tower heights. Line-of-sight distance is a simple Pythagorean Theorem problem - d = square root[(r+h)^2 - r^2], where r is the radius of Earth and h is the height of your tower. This is assuming you are significantly shorter than the tower - thus, your own height is negligible.

If you want to walk there, you need the arc length that is described by the segment of the circle you have cut out with the two radii drawn from the tangent point and the tower. You have created a triangle with these radii (and the line-of-sight distance), with one leg of length r and the hypotenuse of length (r+h). The angle between these is equal to the arc cosine of their ratio - arccos[r/(r+h)]. The length of an arc is equal to the radius times the angle swept through. So, the walking distance is equal to r times arccos[r/(r+h)]. BUT you need to do the trig in radians... or else you stare at a piece of paper with scribbles over it for quite a long time wondering why your answer is two orders of magnitude larger than it should be.

posted by backseatpilot at 6:54 PM on May 4, 2008

The Earth can be thought of as a circle with radius 6378.1 km, or about 20 925 525 feet. If you imagine your tower being at the North Pole, the point you are looking for is where a line drawn from the top of the tower intersects the circle at only one point - the tangent line. If you draw a line from the center of the Earth to this tangent point and another from the center of the Earth to the base of the tower, you have a right triangle; one property of a tangent line is that it is perpendicular to the radius of the circle that goes through the point where it intersects the circle.

Now you need to consider which distance you're looking for - line-of-sight distance from your eye to the top of the tower, or the distance you would need to walk to reach the tower. They're about the same for short tower heights. Line-of-sight distance is a simple Pythagorean Theorem problem - d = square root[(r+h)^2 - r^2], where r is the radius of Earth and h is the height of your tower. This is assuming you are significantly shorter than the tower - thus, your own height is negligible.

If you want to walk there, you need the arc length that is described by the segment of the circle you have cut out with the two radii drawn from the tangent point and the tower. You have created a triangle with these radii (and the line-of-sight distance), with one leg of length r and the hypotenuse of length (r+h). The angle between these is equal to the arc cosine of their ratio - arccos[r/(r+h)]. The length of an arc is equal to the radius times the angle swept through. So, the walking distance is equal to r times arccos[r/(r+h)]. BUT you need to do the trig in radians... or else you stare at a piece of paper with scribbles over it for quite a long time wondering why your answer is two orders of magnitude larger than it should be.

posted by backseatpilot at 6:54 PM on May 4, 2008

12.2 miles. Surprised no one answered this directly.

posted by crapmatic at 7:01 PM on May 4, 2008 [4 favorites]

posted by crapmatic at 7:01 PM on May 4, 2008 [4 favorites]

Although it doesn't get repeated as much nowadays (because it is in fact a 18th century falsification) this type of observation when applied to a ship's mast at sea is one obvious way that the ancients did in fact know that the earth was round, not flat. At sea you see a ship's mast before the hull as it approaches. That wouldn't be the case of the world was flat.

posted by wfrgms at 7:08 PM on May 4, 2008

posted by wfrgms at 7:08 PM on May 4, 2008

*have literally no break in my line-of-sight*

Nevada has some good flat areas. (Those pictures show far more mountains than there actually are. I think it's a phenomenon like "oh look! something to actually photograph!")

posted by salvia at 7:22 PM on May 4, 2008

Hmm, yeah, I was wrong about saying Nevada is flat. "Almost all of Nevada belongs physiographically to the Great Basin, a plateau characterized by isolated mountain ranges separated by arid basins."

I think I confused the idea of flatness with solitude. Here's what I remember about driving through Nevada on Highway 50, "the loneliest highway in America" -- taking bets on how far it was to that thing in the distant horizon, then watching on the speedometer to see how far it was. Seven miles was common. And that thing (on Highway 50) was usually not a building, and we usually made it there without passing another car.

posted by salvia at 7:51 PM on May 4, 2008

I think I confused the idea of flatness with solitude. Here's what I remember about driving through Nevada on Highway 50, "the loneliest highway in America" -- taking bets on how far it was to that thing in the distant horizon, then watching on the speedometer to see how far it was. Seven miles was common. And that thing (on Highway 50) was usually not a building, and we usually made it there without passing another car.

posted by salvia at 7:51 PM on May 4, 2008

Ah, yeah, Nevada does have a lot of solitude. So does Arizona and New Mexico, I believe. Alaska and the Yukon... Montana, Colorado, etc etc etc... The U.S. and Canada have a lot of places you can find solitude.

posted by krisak at 7:56 PM on May 4, 2008

posted by krisak at 7:56 PM on May 4, 2008

One of the more incredible places on the planet is the incredible beauty of the Flint Hills area of Kansas. I could not believe how far one can see. It's like looking past the horizon. The "hills" aren't enough to really notice untill you are atop them and realize that you can all of a sudden see forever. It's a magnificent place.

National Geographic

National Wildlife Refuge

Kansas Flint Hills

posted by Gerard Sorme at 7:59 PM on May 4, 2008 [1 favorite]

National Geographic

National Wildlife Refuge

Kansas Flint Hills

posted by Gerard Sorme at 7:59 PM on May 4, 2008 [1 favorite]

Google Maps has a terrain feature now.

Looks like anywhere down the middle of the country would do for relatively flat topography (rolling hills notwithstanding).

What about the top of a mountain? Or, hell, the top of a building?

posted by Sys Rq at 8:02 PM on May 4, 2008

Looks like anywhere down the middle of the country would do for relatively flat topography (rolling hills notwithstanding).

What about the top of a mountain? Or, hell, the top of a building?

posted by Sys Rq at 8:02 PM on May 4, 2008

*where in America I could go and have literally no break in my line-of-sight*

Mauna Kea, Hawaii

It's the highest point in Hawaii. And, of course, it's on an island surrounded by water. No pesky hills on the oceans to block your sight.

posted by Cool Papa Bell at 8:29 PM on May 4, 2008

*At sea you see a ship's mast before the hull as it approaches. That wouldn't be the case of the world was flat.*

I don't think this would have convinced me. Mirages do all sorts of weird things to light on the horizon--maybe hiding ship hulls was just a similar phenomenon.

P.S. I do, in fact, know that the world is round--I'm just saying back in the day.

posted by IvyMike at 10:06 PM on May 4, 2008

I remember that on the causeway across Lake Ponchetrain there's a point where you can't see land (or buildings) in any direction. There's just you, the highway, and water. It's rather a peculiar feeling.

posted by Class Goat at 10:53 PM on May 4, 2008

posted by Class Goat at 10:53 PM on May 4, 2008

krisak, I counter with:

North Dakota (eastern end)

The same lonely road thing. The only thing on horizon is the next sugar beet processing plant. Not as vast as other areas, but the one hour stretch of perfectly straight road... I've seen it waay too many times.

posted by easyasy3k at 12:51 AM on May 5, 2008

North Dakota (eastern end)

The same lonely road thing. The only thing on horizon is the next sugar beet processing plant. Not as vast as other areas, but the one hour stretch of perfectly straight road... I've seen it waay too many times.

posted by easyasy3k at 12:51 AM on May 5, 2008

Driving through Buffalo Gap Grasslands in South Dakota, there were many places where there was, effectively, nothing on the horizon. It is surprisingly eerie.

posted by dzot at 6:15 AM on May 5, 2008

posted by dzot at 6:15 AM on May 5, 2008

This thread is closed to new comments.

Working on some math homework?

posted by krisak at 6:13 PM on May 4, 2008