Earth Curvature Calculator (hidden height & drop over distance)

Understanding the Earth curvature calculator — concept and principle

This calculator works out how much of a distant object is hidden behind the curve of the Earth, how far the surface drops below a level line, and how far away your own horizon is. People use it to check whether a lighthouse, ship, island or skyline should be visible across water, and to understand photos taken over long distances.

From 2 m eye height, a 50 m tower 20 km away loses its bottom 14.5 m behind the bulge.

How it is calculated

Horizon distance s1 = R × arccos(R ÷ (R + h))

Hidden height = R ÷ cos((s − s1) ÷ R) − R (zero if s ≤ s1)

Drop below a level line = R ÷ cos(s ÷ R) − R ≈ s² ÷ 2R

h is your eye height in metres, s the surface distance to the object in km, and the object height sets the visible and hidden share.
R = 6,371 km. With standard refraction R is replaced by an effective radius R ÷ (1 − 0.13), about 7,323 km.

The principle

Your line of sight just grazes the Earth at the horizon. Everything beyond that point sits lower than your sight line, and the bottom of a distant object is cut off by the bulge between the horizon and the object. Air bends light slightly downward, following the curve, so objects look a little higher than pure geometry predicts. Using a larger effective radius is the standard way to include this average bending; k = 0.13 reproduces the navigator's horizon rule.

Worked example

Default inputs: distance 20 km, eye height 2 m, object height 50 m, standard refraction.

Effective radius R = 6,371,000 ÷ 0.87 = 7,322,989 m.
Horizon: s1 = R × arccos(R ÷ (R + 2)) = 5,412 m = 5.41 km.
Hidden height = R ÷ cos((20,000 − 5,412) ÷ R) − R = 14.5 m.
Visible height = 50 − 14.5 = 35.5 m, so 29.1% of the object is hidden.
Drop below a level line at 20 km = 27.3 m.

Each point is the calculator re-run with only “Distance to the object (km)” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Distance to the object (km)	Hidden height of the object (m)
5	0
10	1.4
15	6.3
20	14.5
25	26.2
30	41.3
40	81.7

How to read the result

If the hidden height is larger than the object, it is entirely below your horizon. Raising your eye height pushes the horizon further away and reduces the hidden part much more than moving closer by the same amount. Without refraction the drop is about 7.85 cm at 1 km and 7.85 m at 10 km, growing with the square of distance.

Common mistakes

Using the drop below a level line as the hidden height. The hidden height starts from your horizon, not from your feet.
Ignoring eye height. Even 2 m moves the horizon about 5 km away.
Expecting the result to hold on every day. Temperature layers over water can bend light much more or less than average.

Keep in mind: Refraction near the surface varies with weather, so mirages and looming can make objects appear or vanish against these figures. Treat the result as a typical value.

Frequently asked questions

How far is the horizon from a beach?

With eyes about 1.7 m above the water the horizon is roughly 5 km away under standard refraction.

Why can I sometimes see a coast that should be hidden?

Strong temperature inversions over water bend light along the curve more than usual, lifting distant objects into view.

Is the 8 inches per mile squared rule correct?

It approximates the drop below a level line, not the hidden height seen from a raised eye, and it ignores refraction.

Sources: NASA NSSDCA, Earth Fact Sheet (mean radius 6,371 km) · NGA Pub. No. 9, The American Practical Navigator (Bowditch), Article 2202

Checked 9 Nov 2026. Spherical geometry with the NASA NSSDCA Earth Fact Sheet mean radius (6,371.0 km, 2024); refraction coefficient k = 0.13 chosen to reproduce the visible-horizon rule of NGA Pub. No. 9, The American Practical Navigator (Bowditch), Article 2202 (D = 2.07 √h nmi).

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