How Far Away Is the Horizon? Calculate in Miles or Kilometers

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Get exact distances or quick estimations with simple math
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Well, the distance to the horizon depends almost completely on how high above sea-level you’re looking from—but, there are a few simple calculations you can use to figure out just how far from the horizon you are. Follow the step-by-step guide below to calculate the distance to the horizon from wherever you’re standing.

Calculating the Distance to the Horizon

Use the formula Distance = 1.22459 x √(height above sea level). Add your height and current elevation above sea level in feet. Take the square root of that sum and multiply it by 1.22459 to get your result in miles. At sea level, divide your height in feet by 2 to get an estimation in miles.

Section 1 of 5:

Calculating the Distance to the Horizon with an Easy Formula

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  1. If you know your height, you can use this measurement as an estimate. Otherwise, use a tape measure to measure the distance between your eyes and the surface you’re standing on. [1]
    • Quick estimation from sea level: If you’re standing on the ocean shore (i.e., at zero sea level), just divide your height in feet by half to estimate the distance to the horizon. If you’re 5' 6" (5.5 feet) tall, for example, the distance to the horizon is about 2.75 miles (5.5 ÷ 2 = 2.75).
  2. Use this calculator to determine the elevation of your current location above sea level. Add additional measurements if you’re standing on a hill, building, boat, or other structure. Sum up these numbers and add them to your result from step 1. This sum is now the value known as h . [2]
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  3. Multiply the value of h by 1.5 (or by 13 if you took the measurements in meters). Then take the square root of that product to get your final result. [3]
    • If you did your calculations in meters, then your final answer will be in kilometers. If you did your calculations in feet, then your final answer will be in miles .
    • The result of this formula is the rough estimate of the distance between you and the horizon. For a more precise distance (if you have a calculator handy), use the formula Distance = 1.22459 x √( h ) to get your result in miles.
    • This formula gives you the distance from your eyes to the horizon if you were to travel in a straight line. The actual distance you'd have to travel to get to the horizon will be longer because of surface curvature and irregularities (on land).
    • If you want to know the actual travel distance to the horizon, proceed to the next section.
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Section 2 of 5:

Calculating the Distance to the Horizon Using the Pythagorean Theorem

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  1. Start with your measured height in meters. Then, add your current elevation to your height to get h .
  2. In most cases, you’ll be calculating the distance to the horizon on the planet Earth. Since we know that the radius of the Earth is approximately 6,371,008 meters, your value r = 6,371,008. [4] Add together r and h to get segment r + h .
  3. Envisioning a circle to represent the Earth, draw the r + h segment from the center of the planet to the point from which the observer (you) is looking. Then, draw a tangent line between the observer’s point of view and the surface of the planet. [5]
    • Finally, complete the triangle by connecting the point where the tangent touches the planet to the center of the globe.
    • This last segment should have the length r and form a right triangle.
  4. We now know that the cathetus of the triangle (a side of a right triangle adjacent to the right angle) has length r and the hypotenuse has length r + h . To find the length of the remaining side d , use the formula d = √(( r+h ) 2 – r 2 ). [6]
    • The value d will give the distance in meters between your viewing point and the horizon.
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Section 3 of 5:

Examples Problems: Calculating Horizon Distance

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  1. Calculate the distance to the horizon if you are 1.68 m (5.51 ft) tall and are standing on flat ground in Surrey, England.
    • The elevation in Surrey, England is 104 m.
    • 104 + 1.68 = 105.68 m = h .
    • Distance = √(13 x 105.68 ) = √(1374.84) = 37.08 kilometers.
  2. Calculate the distance to the horizon if you are 1.68 m (5.51 ft) tall and are standing on flat ground in Surrey, England.
    • The elevation in Surrey, England is 341 ft.
    • 341 + 5.51 = 346.51 feet = h .
    • Distance = 1.22459 x √( 346.51 ) = 1.22459 x 18.61 = 22.8 miles.
  3. Calculate the distance to the horizon if you are 1.68 m (5.51 ft) tall and are standing on flat ground in Surrey, England.
    • h = 1.68 m.
    • r = 6,371,008 m.
    • r + h = 6,371,009.68 m.
    • Distance ( d ) = √(( 6,371,009.68 ) 2 – 6,371,008 2 ) = √(4.0589764e+13 – 4.0589743e+13) = √(21,000,000) = 4582.58 meters = 4.58 kilometers.
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Section 4 of 5:

What is the horizon?

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  1. The horizon can only be observed if viewed from close to the surface of a planet or celestial body. As a human on planet Earth, you’re standing on the Earth’s surface and seeing the curvature of the globe as it meets the sky—that makes the horizon line. [7]
    • The horizon is a changing thing—its appearance and properties are influenced by tis surrounded environment.
    • The horizon line is only well-defined in mathematics if there are no obstacles in your line of view. For this reason, the best way to see or calculate the horizon is to go to the seashore and face the open sea.
Section 5 of 5:

Frequently Asked Questions About Horizon Distance

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  1. The average person’s range of vision is infinite and able to see for miles and miles. However, you can only see up to 3 miles (4.8 km) on Earth before you encounter the horizon, which is caused by the curvature of the Earth. However, you can see objects like mountains and skyscrapers that are further away because there are no horizons to obstruct them. [8]
    • On a clear night, you can see stars that are millions of miles away.
    • Of course, any given person’s range of vision may be affected by personal factors such as eye health and the brightness and size of the object they’re looking at.
  2. The furthest you can see while on the ocean (or at sea level) is about 2.9 miles (4.7 km). This distance is limited by the horizon and the Earth’s natural curvature. [9]
  3. Since the Moon is smaller than Earth, the horizon would be closer to you on the Moon than if you were on Earth. To calculate the distance to the horizon on the Moon, add together the radius of the Moon and your height in meters. Square that sum, then subtract it by the square of the Moon’s radius. Take the square root of this final result to get your distance in meters.
    • For someone who’s 1.75 metres (5.7 ft) tall, for example, the horizon on the Moon would be about 2.5 kilometres (1.6 mi) away. [10]
  4. If you were the average human height 1.75 m (5.7 ft) and stood on Earth with an unobstructed view of the horizon, the distance between you and the horizon would be approximately 4.7 kilometres (2.9 mi).
  5. To see 10 kilometres (6.2 mi) away, you only need to be 7.85 metres (25.8 ft) above the ground.
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  • Question
    Can I see an object beyond the horizon?
    Community Answer
    Not the entire object, but you might be able to see part of it if it's tall enough. If something like a tree or a building is standing just beyond the horizon, you might be able to see the upper part of it.
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  • Question
    Why wouldn't the formula for determining distance to the horizon work in a city or a forest?
    Community Answer
    The trees in the forest or the buildings in the city would affect the view.
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  • Question
    How many acres in 630 sq ft?
    Donagan
    Top Answerer
    Because there are 43,560 square feet in an acre, 630 square feet equal 630 / 43,560 = 0.01446 acre.
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      References

      1. ↑ http://www.fao.org/fishery/static/FAO_Training/FAO_Training/General/x6707e/x6707e04.htm
      2. ↑ http://www.fao.org/fishery/static/FAO_Training/FAO_Training/General/x6707e/x6707e04.htm
      3. ↑ https://www.sciencefocus.com/planet-earth/how-far-away-is-the-horizon
      4. ↑ https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html
      5. ↑ https://spacemath.gsfc.nasa.gov/Insight/Insight16.pdf
      6. ↑ https://spacemath.gsfc.nasa.gov/Insight/Insight16.pdf
      7. ↑ https://education.nationalgeographic.org/resource/horizon/
      8. ↑ https://optical-academy.com/blog/how-far-can-the-human-eye-see/
      9. ↑ https://time.com/archive/6766128/science-how-far-can-you-see/
      1. ↑ https://www.spaceacademy.net.au/library/notes/horizons.htm

      About This Article

      Article Summary X

      Before calculating distance to the horizon if you’re standing exactly at sea level, start by measuring your total height, unless you already know it. Once you’ve got your height, measure the distance between the ground and your eyes. Then, subtract the distance measurement from your total height. The amount remaining is the distance between your eyes and the surface you’re standing on. Multiply by 13 meters if you took the measurement in meters, or multiply by 1.5 feet if you took the measurement in feet. Take the square root to get your answer, which will be in kilometers if you measured in meters or in miles if you measured in feet. Your calculated distance is a straight line from your eyes to the horizon. For more tips, including how to measure from an elevated point, read on.

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