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Earth Curvature Calculator

Table of contents

What is the Earth's curvature?Curvature of Earth per mileHow far can I see before the Earth curves?Calculating the obstructed height of an objectIs this Earth curvature calculator accurate?FAQs

This Earth curvature calculator allows you to determine how much of a distant object is obscured by the Earth's curvature. So, if you ever wanted to estimate the total height of a target that is partially hidden behind the horizon, now you can. You will also be able to find out how far you can see before the Earth curves – that is, what is your distance to the horizon.

Don't worry if you don't know what the curvature of Earth is yet – simply keep reading to learn all the necessary information!

What is the Earth's curvature?

Imagine you are looking at the sea. There is no land in sight, only the endless blue waters shimmering in the afternoon sun. You can make out the line that divides the sea and the sky. This line is called the horizon.

Suddenly, you begin to see a point that is getting larger and larger. First, it is a top of a white sail; when it moves closer, you can also notice the shape of a ship. Where was this ship before? It was hidden behind the horizon.

The reason for this is obvious: as Earth's shape is very similar to a sphere – the surface between you and the ship is not entirely flat but "bulges" up a bit. That is why it has obstructed your view. The curvature of Earth is simply the measure of this "bulge". It is expressed as the height of the "bulge" per kilometer or mile.

💡 Note that if the Earth were flat, you would see the whole ship even from afar. Expect it would start out very small whole ship (a dot) and then get larger as it got closer. However, by observing just the top of the sail from afar first, you've seen for yourself that the Earth is not flat. If you are interested in exploring this further, you should check out our flat vs. round earth calculator.

Curvature of Earth per mile

How large is the curvature of Earth, then? As we don't notice it in our everyday lives, it has to be relatively small. Most sources consider 8 inches per mile as the most accurate estimate. That means that for every mile between you and an object, the curvature will obstruct 8 inches of the object's height.

How far can I see before the Earth curves?

The first thing you can find with our Earth curvature calculator is the exact distance between you and the horizon. You only need to know two values: your eyesight level (in other words, the distance between your eyes and mean sea level – assuming you are looking out to sea) and the radius of the Earth. Input these numbers into the following equation:

a=(r+h)2r2a = \sqrt{(r + h)^2 - r^2}

where:

  • aa — Distance to the horizon;
  • hh — Eyesight level above mean sea level; and
  • rr — Earth's radius, equal to 3959 miles or 6371 km.

This equation can be derived using the Pythagorean theorem. You can try to derive it yourself – it is not that hard!

Calculating the obstructed height of an object

The estimation of an obscured height from the observer perspective at a curved surface.

Look at the image above. It represents a situation analogical to the one with the ship from above. You can see a part of the object, but the rest of it is hidden behind the horizon. If you want to know the height of the obstructed thing, simply enter all the necessary values into the Earth curvature calculator. You can also calculate the height manually:

  1. Determine the distance between you (the observer) and the lowest point of the object that you can actually see. Let's call this value dd and assume it is equal to 25 miles.

  2. Measure your eyesight level – that is, the height at which your eyes are above the sea. We will denote it with the letter hh. We can assume that it is equal to 6 feet, which is approximately 0.0011364 miles.

  3. Calculate the distance between you and the horizon, aa, using the formula mentioned above:

a= (r+h)2r2=(3959+0.0011364)239592= 3 miles\qquad\footnotesize \begin{split} a =&\ \sqrt{(r + h)^2 - r^2} = \\[0.5em] &\sqrt{\! (3959 \! + \! 0.0011364)^2 \! - \! 3959^2} \\[0.5em] =&\ 3 \mathrm{\ miles} \end{split}
  1. Now, you can input these values into the second formula to find the height of the obstructed part of the object xx:
x=a22ad+d2+r2rx=322×3×25+252+395923959x=0.0611 miles=322.76 ft\qquad\footnotesize x \! = \! \! \sqrt{a^2 - 2ad + d^2 + r^2} - r \\[1em] \qquad x \! = \! \! \sqrt{\! 3^2 \! - \! 2 \! \times \! 3 \! \times \! 25 \! + \! 25^2 \! + \! 3959^2} \\[0.5em] \qquad\qquad - 3959 \\[1em] \qquad x \! = 0.0611 \mathrm{ \ miles} = 322.76 \mathrm{\ ft}
  1. If you have trouble with units conversion, simply use our length converter.

Is this Earth curvature calculator accurate?

You might find that if you were to test our calculator vs. a real-life scenario, our calculator might be slightly wrong in some cases. Why does this happen? Does it mean the Earth is flat and doesn't curve at all?

Of course not! It just means that our calculator doesn't account for the phenomenon of refraction. When light travels through a medium that is not perfectly uniform, such as air, it bends or refracts. For example, refraction can happen when light hits a pocket of cold air or a hot draft of rising air.

As the ray of light bends slightly, it changes direction. That means that some photons from the object that would usually hit the ground can bend around the Earth's surface and reach your eye, so the heights and distances as shown in the picture above may seem to be different. That's why, when you are calculating the obstructed height of an object, the distances you see maybe a bit different from the observed ones!

If you are interested in reading more about snell's law of refraction, you should check out our Snell's law calculator here.

FAQs

How far is the horizon at sea level?

The horizon at sea level is approximately 4.5 km. To calculate it, follow these steps:

  1. Assume the height of your eyes to be h = 1.6 m.

  2. Build a right triangle with hypotenuse r + h (where r is Earth's radius) and a cathetus r.

  3. Calculate the last cathetus with Pythagora's theorem: the result is the distance to the horizon:

    a = √[(r + h)² - r²]

  4. Substitute the values in the formula above:

    a = √[(6,371,000 + 1.6)² - 6,371,000²] = 4,515 m

How do I calculate the distance of the horizon?

To calculate the theoretical distance of the horizon from your point of view, imagine building a right triangle with sides equal to:

  • Earth's radius plus the height of your eyes above sea level, r + h;
  • Earth's radius r; and
  • The line tangent to the surface of the Earth starting in your eyes. That's the distance of the horizon.

We calculate the distance of the horizon with the formula a = √[(r + h)² - r²].

Can you see France from England?

Yes, but only in excellent conditions. From the Cliffs of Dover, with a height of about 100 m, your horizon would be at 35.7 km. You can calculate this distance with this formula, where h is your elevation and r is Earth's radius:

a = √[(r + h)² - r²]

Since the narrowest part of the Channel is barely 33 km, you can see France but by a hair's breadth.

From how far can you see Mount Everest?

You can see Mount Everest (theoretically) from a distance of 340 km.

Assuming you are at sea level, with your eyes at 1.6 m above the ground, your horizon (at a distance of about 4.5 km) would cover the entirety of the tallest mountain on Earth only if you are at more than 340 km away from it. Theoretically, you can see the summit of Everest from Bangladesh; however, other peaks cover it!

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