Earth Curvature Calculator

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Previous page : Eight Inches per Mile squared
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A calculator to calculate the hidden height, etc.

On the previous page, we had a look at the problem with the “eight inches per miles squared” formula. We will here look at a much, much better way to calculate the hidden height as one looks out over the sea.

To the Curvature calculator

In the calculator, you should enter:

The observer height, or altitude, i.e. the height above the water level of the observer’s eye or camera lens.
The distance to the observed object.

The values can be entered using metric or imperial units. You can enter the values with the corresponding Enter buttons or the Enter key.

The coefficient of refraction is already there, but you can change the value if you wish. The value, 0.13, is quite often used as a default value. That corresponds to an apparent radius of the Earth that is about 15% larger than the actual size.

The Earth radius used is the average radius.

You can then press the Calculate button – or press the letter C. It will now calculate the hidden height + numerous other values, like the dip angle to the horizon.

If the object is in front of the horizon, the hidden height will be given as a negative value. This value shows how much of the object is in front of, and lower than, the horizon.

The calculator can do more. We will look at that on the next page.

The derivation of the formulas

Let R be the radius of the Earth, h the observer altitude, H the hidden height, d the distance to the horizon, and D the distance from the horizon to the top of the hidden height.

We have two straight angle triangles, OAB and OBC.

We get two equations using the Pythagorean theorem. `

$\displaystyle \left\{ \begin{array}{l}{{\left( {R+h} \right)}^{2}}={{R}^{2}}+{{d}^{2}}\\{{\left( {R+H} \right)}^{2}}={{R}^{2}}+{{D}^{2}}\end{array} \right.$

We can expand the first to get

$\displaystyle {{R}^{2}}+2Rh+{{h}^{2}}={{R}^{2}}+{{d}^{2}}$

$\displaystyle d=\sqrt{{2Rh+{{h}^{2}}}}$

We have that the distance from A to B, let us call it L is L=d+D or D=L–d. We can substitute that into the second Pythagorean equation to get

$\displaystyle {{\left( {R+H} \right)}^{2}}={{R}^{2}}+{{\left( {L-d} \right)}^{2}}$

That gives us

$\displaystyle H=\sqrt{{{{R}^{2}}+{{{\left( {L-d} \right)}}^{2}}}}-R$

Then we need to consider refraction. The refraction bends the light downwards, causing light from further away to reach us. This will cause the Earth to look slightly flatter than it is, and thus it will look like it has a slightly larger radius than it has. The amount of refraction is denoted by the coefficient of refraction, k. And the apparent radius is calculated as

$\displaystyle R=\frac{{{{R}_{0}}}}{{1-k}}$

where R₀ is the actual radius and R is the apparent radius. The value for k varies with the atmospheric condition, but k=0.17 is a commonly used value.

So, to find the hidden height, we first calculate the refracted radius, then the distance to the horizon, then, finally the hidden height H.

The dip angle and the length along the ground are also calculated.

If you want to find the drop hight you can set the higt of the observet to 0.

To the Curvature calculator

Up a level : The Shape of the Earth
Previous page : Eight Inches per Mile squared
Next page : Earth Radius Calculator

Last modified: May 4, 2025 @ 09:45