Have you ever looked at a sunset and asked, "How far am I from the horizon?" If you know your eye level from sea level, you can calculate the distance between you and the horizon.
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Method 1 of 3: Measuring Distances with Geometry
Step 1. Measure "eye height
Measure the distance between the eyes and the ground (use meters). One easy way is to measure the distance from the crown to the eye. Then, subtract your height from the distance between the eyes and the crown that you have measured. If you standing right at sea level, then the formula is as follows.
Step 2. Add your "local elevation" if it stands above sea level
How high is your standing position from the horizon? Add that distance to your eye level (return to meters).
Step 3. Multiply by 13 m, because we are counting in meters
Step 4. Square root of the result to get the answer
Since the unit used is meters, the answer is in kilometers. The calculated distance is the length of a straight line from the eye to the horizon point.
The actual distance will be longer due to the curvature of the earth's surface and other abnormalities. Continue to the next method for a more accurate answer
Step 5. Understand how this formula works
This formula is based on a triangle formed by the point of observation (that is, both eyes), the point of the horizon (which you see), and the center of the earth.
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By knowing the radius of the Earth and measuring eye height plus local elevation, only the distance from the eye to the horizon remains unknown. Since the two sides of the triangle that meet at the horizon form an angle, we can use the Pythagorean formula (formula a2 + b2 = c2 classical) as the basis for calculations, namely:
• a = R (Earth radius)
• b = distance to horizon, unknown
• c = h (height of the eye) + R
Method 2 of 3: Calculating Distance Using Trigonometry
Step 1. Measure the actual distance you have to travel to reach the horizon with the following formula
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d = R * arccos(R/(R + h)), where
• d = distance to horizon
• R = Earth's radius
• h = eye height
Step 2. Increase R by 20% to compensate for light refraction distortion and get an accurate answer
The geometric horizon calculated by this method may not be the same as the optical horizon seen by the eye. Why?
- The atmosphere bends (refracts) light traveling horizontally. This means that light can slightly follow the curve of the earth so that the optical horizon appears further away from the geometric horizon.
- Unfortunately, refraction due to the atmosphere is neither constant nor predictable due to changes in temperature with altitude. Therefore, there is no simple way to correct the formula for the geometric horizon. However, there is also a way to obtain an "average" correction by assuming the earth's radius is slightly larger than the original radius.
Step 3. Understand how this formula works
This formula calculates the length of the curved line that runs from your feet to the original horizon (marked in green in the image). Now, the arccos portion (R/(R+h)) refers to the angle at the center of the earth formed by the line from your feet to the center of the earth and the line from the horizon to the center of the earth. This angle is then multiplied by R to get the "length of the curve," which is the answer you're looking for.
Method 3 of 3: Alternative Geometric Formulas
Step 1. Imagine a flat plane or ocean
This method is a simplified version of the first set of instructions in this article. This formula only applies to feet or miles.
Step 2. Find the answer by entering the eye height in the formula in feet (h)
The formula used is d = 1.2246* SQRT(h)
Step 3. Derive the Pythagorean formula
(R+h)2 = R2 + d2. Find the value of h (assuming R>>h and the radius of the earth is shown in miles, approximately 3959) then we get: d = SQRT(2*R*h)
