In the table, the first column is your height in meters above the Earth’s surface (really the height of your eyes) and the second column is the horizon distance in kilometers.
I put this into an Excel spreadsheet, and the numbers are below. We can use this to put in different values for h, our height, and see how far away the edge of the Earth is. So now we have an equation that tells us how far away the horizon is depending on where we are above the surface. Now, take the square root of both sides, and voila! You get d. Hey, we have a factor of R2 on both sides, so they cancel! That leaves us with: Substitute that back into the first equation to get Now what? Well, let’s expand that last term using FOIL : This gives us the following algebraic formula:
#Last horizon cool math plus
One side is d, the other is R, and the hypotenuse is the Earth’s radius plus your height above the surface, R + h. The square of the hypotenuse is equal to the sum of the squares of the other two sides. That means we have a right triangle, and - reach back into the dim, dusty memory of high school - that means we can use the Pythagorean Theorem to get d. The key thing here is that at the visible horizon, the angle between your line-of-sight and the radius line of the Earth is a right angle (marked in the diagram). Note that the radius of the Earth is a constant, but that d will vary as h goes up or down. The line-of-sight to the horizon is the red line, labeled d. The dude standing on the Earth is a human of height h (not to scale, huge duh there). The Earth’s radius varies with latitude, but I’ll just use 6365 kilometers as a decent average. In this diagram, the circle is the surface of the Earth, which has a radius of R. What does our situation look like? Well, it looks something like this: Let’s assume the Earth is a perfect sphere, because that makes things a lot easier. All it takes is a little knowledge of geometry, and a diagram to show you the way.įollow along with me here. But in fact it’s a real thing, and the distance to it can be determined. The horizon is a semi-mythical distance, used in poetry as a metaphor for a philosophical division of some kind. But even then, the landscape blows past, and eventually you wind up flying over eastern Colorado, and there’s nothing to see but flat, flat land, extending all the way to the horizon.Īnd as I gaze over the amber waves of grain to the line that divides land and sky, I sometimes wonder how far away that line is. However, sometimes I do like to grab a window seat, especially if I’m flying near sunset, or over a particularly interesting landscape (flying over southern Utah near sunset will change your life). I usually prefer an aisle seat, because then the rude guy who smells funny and spreads over 1.8 seats only irritates me on one side, and I’m not wedged up against the window.
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