Curvature Calculator + Earth Curvature Formula

Surveyors and maritime businesses deal with measurements involving the Earth’s curvature. Using formulas and surveying instruments, they can determine the part of a distant object which gets obscured by this curvature. This allows them to come up with a good estimate of the total height of an object that’s partially hidden by the horizon. In such a case, this curvature calculator comes in handy for quick calculations.

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How to use the curvature calculator?

This curvature calculator is as simple as they come. As long as you have the required values, you can use this online tool without having to calculate by hand using the Earth curvature formula. Here are the steps to follow:

  • First, enter the value of the Distance to the Object and choose the unit of measurement from the drop-down menu.
  • Then enter the value of the Eyesight Level and choose the unit of measurement from the drop-down menu.
  • After entering both values, the Earth curvature calculator automatically generates for you the value of the Distance to Horizon and the value of the Obscured Object Part.

What is the curvature of the Earth?

Imagine yourself looking far at an open sea where you see no land, only the seemingly endless blue water. On clear days, you may see the horizon dividing the sky and the sea. Maybe by chance, you would also see a certain point on the horizon that starts to get bigger and bigger.

As it draws closer, you discover that it’s a ship and the first thing you would notice is its sail. The closer it gets, the more the shape of the ship becomes distinct. Before you can make such a sighting, you can ask yourself the questions, “Where was the ship before it made its gradual appearance?” and “Was it concealed by the horizon?”

At this point, we already know that the Earth is a sphere and not flat. Therefore, between the ship and yourself, it does bulge up quite a bit. This is the reason why it obstructed you from the view. This “bulge” is what’s known as the Earth’s curvature. You measure this value as the height of the “bulge” per mile or kilometer.

How to calculate the curvature of the Earth?

The degree of obstruction of a distant object caused by the curvature of the Earth is dependent on several factors namely:

  • The object’s distance.
  • The observer’s height.
  • The object’s height.
  • The magnitude of atmospheric refraction.

“Flat-Earthers” or those people who still believe that the Earth is flat, prefer using the visibility of faraway objects so they can prove that the curvature of the Earth is just a myth. Of course, as you would expect, their theories fail because they don’t take into account the observer’s height and the atmospheric refraction.

Aside from these factors, they also make a few more mistakes like errors in unit conversion, errors in distance calculation, and so on. If only these flat-Earthers gave these due considerations and they tried to fix their mistakes, everything should turn out differently. They would then discover that we have a spherical Earth.

Not taking into account the observer’s height is the most common mistake when calculating the Earth’s curvature whether you use a curvature calculator or the Earth curvature formula. All that is usually done is to calculate the drop from a horizontal plane.

Even if you use a professional-grade AutoCAD software which is highly precise, the resulting numbers will still turn out incorrect if you made a mistake in the geometry in the first place.

Not accounting for atmospheric refraction is another very common mistake. Atmospheric refraction refers to the deviation of light from a straight line as it passes through the atmosphere due to the variation in air density as a function of height.

In simpler terms, atmospheric refraction usually bends light to follow the Earth’s curvature to a specific point. This causes an object to appear higher above the horizon than it actually is. It’s also important to note that atmospheric refraction is never a constant as it would depend on weather conditions and how much refraction may vary in one day.

If you have taken all of this information into consideration and you have corrected all mistakes, you will be able to find the values you need through a simple process. Here is how to calculate for the Earth’s curvature without using a curvature calculator:

For this example, let’s assume that the Earth has a spherical shape with a radius of 3,963 miles. If you’re at point P on the surface of the Earth, then move tangent to the surface a distance of one mile, in order to form a right triangle. Then using the Pythagorean theorem:

c² = a² + b²
c² = (3963)² + (1)²
c² = 15705370

Then take the square root of both sides of the equation:

c = 3963.000126 miles

You should also solve for your position above the Earth’s surface:

P = 3963.000126 – 3963
P = 0.000126 miles
P = 7.98 inches

This result means that the surface of the Earth curves at approximately 8 inches for every mile.

How much does the Earth curve per mile?

The Earth does have a curvature but almost nobody ever notices it because the value is very small. For those in the know, the most precise value of curvature is about 8 inches per mile. This means that for each mile of distance between you and another object, the curvature obstructs approximately 8 inches of the object’s height.

How far can you see Earth curvature?

The Earth curvature calculator yields the distance between yourself and the horizon. There are only two values needed to solve this, namely the level of your eyesight or the distance between the ground and your eyes and the Earth’s radius. Enter these values into the curvature equation:

a = √[(r + h)² – r²]

a refers to the distance to the horizon
h refers to the level of your eyesight
r refers to the radius of the Earth which is 3963 miles

The equation itself comes from the Pythagorean Theorem. You can even try deriving the equation yourself, it isn’t that difficult!