Have you ever watched the sun disappear on the horizon wondering "How far is the horizon from where I am?" If you can measure the height of your eyes with respect to sea level, you can actually calculate the distance between you and the horizon as explained below.
Steps
Method 1 of 3: Calculate the distance using geometry
Step 1. Measure "the height of your eyes"
Measure the length between your eyes and the ground in meters or feet. One way to calculate this is to measure the distance between your eyes and the tip of your head. Subtract this value from your total height and what will remain is the distance between your eyes and the surface you are standing on. If you are exactly at sea level, with the soles of your feet at water level, this will be the only measure you need.
Step 2. Add your “local elevation” if you are on a high surface, such as a hill, a building or a boat
How many meters above the true horizon line are you? A meter? 4000 feet? Add this value to the height of your eyes (obviously using the same unit of measurement).
Step 3. Multiply by 13m if you measured in meters, or by 1.5ft if you measured in feet
Step 4. Calculate the square root to get the result
If you used meters, the result will be in kilometers, if you used feet it will be in miles. The calculated distance is the line between your eyes and the horizon.
The real distance to travel to reach the horizon will be longer due to the earth's curvature or (on land) irregularities. Move on to the method below for a more accurate (but more complicated) formula
Step 5. Understand how this calculation works
It is based on the triangle formed by: your observation point (your eyes), the real point of the horizon (the one you are looking at) and the center of the Earth.
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Knowing the radius of the Earth and measuring the height of your eyes at the local altitude, only the distance between your eyes and the horizon remains as an unknown. Since the sides of the triangle that meet on the horizon actually form a right angle, we can use the Pythagorean Theorem (the good old2 + b2 = c2) as a basis for the calculation, where:
• a = Ra (radius of the Earth)
• b = distance of the horizon, unknown
• c = h (height of your eyes) + R
Method 2 of 3: Calculate the distance using trigonometry
Step 1. Calculate the real distance to cross to reach the horizon line using the following formula
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d = R * arccos (R / (R + h)), where
• d = distance of the horizon
• R = radius of the Earth
• h = eye height
Step 2. Increase the R value by 20% to compensate for the distorted refraction of the light rays and obtain a more accurate measurement
The geometric horizon calculated using the method in this article may not be the same as the optical horizon, which would be what you really see. For what reason?
- The atmosphere distorts (refracts) the light that travels in a straight line. This, in fact, means that the rays of light can slightly follow the Earth's curvature, so the optical horizon is further away than the geometric horizon.
- Unfortunately, atmospheric refraction is neither constant nor predictable, depending on the change in temperature with altitude. So there is no simple method to add a correction to the formula for the geometric horizon, although an "average" correction can be obtained by assuming the earth's radius slightly longer than the real radius.
Step 3. Understand how this calculation works
This will measure the length of the curve that joins your feet to the real horizon (in green in the image). Now, the quantity arccos (R / (R + h)) refers to the angle at the center of the Earth formed by the line that joins the horizon to the center and the line that goes from you to the center. Once we have found this angle, we multiply it by R to find the "length of the arc" which, in this case, is the distance you are looking for.
Method 3 of 3: Alternative geometric calculation
Step 1. Consider a flat surface or the ocean
This method is the simplified version of the first set of instructions shown in this article, and only applies in miles and feet.
Step 2. Find the distance in miles by entering the height of your eyes (h) expressed in feet in the formula
The formula you will be using is d = 1.2246 * SQRT (h)
Step 3. Get the formula from the Pythagorean Theorem
(R + h)2 = R2 + d2. Finding h (assuming R >> h and expressing the radius of the Earth in miles, about 3959), obtains the expression d = SQRT (2 * R * h)
