Earth Curvature Calculator
See how Earth curvature impacts visibility by entering object distance and eye level. The tool estimates hidden height and horizon distance.
Earth Curvature Calculator
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What the Earth curvature calculator does
The Earth curves away beneath your line of sight, hiding the lower parts of distant objects. This calculator works out two things: how far away your horizon is for a given eye height, and how much of a distant object is hidden below that horizon by the curve of the planet.
Below is what Earth's curvature hides, the equations behind it, why ships disappear from the bottom up, and a worked example.
How to use it
- Choose what to calculate: your horizon distance from eye level, or your eye level from a horizon distance.
- Enter the distance to the object and your eye height, or the horizon distance.
- Press Calculate for the horizon distance and the hidden height, or Reset to clear them.
What Earth's curvature hides
Because the Earth is round, its surface curves away from any straight line of sight. Stand looking out to sea, and the water bulges up slightly between you and a distant point, blocking the lower part of anything beyond the horizon. A faraway ship, building, or mountain is not seen in full; its base is hidden behind the bulge of the planet, and only the upper part rises into view. The farther away the object, the more of it is concealed.
This effect is the most everyday evidence that the Earth is a sphere rather than flat. On a flat Earth, distant objects would simply shrink with distance while staying fully visible from base to top. Instead, they are progressively swallowed from the bottom up, exactly as a curved surface predicts. This calculator quantifies the effect, telling you both how far your horizon extends and how much of a distant object the curvature hides from view.
The equations it uses
The calculator uses two pieces of spherical geometry. The horizon distance from an eye height comes from the same right-triangle relationship that governs any horizon:
dhorizon = √((R + h)² − R²)
where R is Earth's radius and h is your eye height. The hidden height of a distant object then follows from how far it lies beyond your horizon. The portion concealed by the curve is the amount the surface drops below your line of sight over that extra distance, which the calculator computes from the object's distance, your horizon distance, and the Earth's radius. Together these tell you where the horizon falls and how much of a distant target sits below it.
Why ships vanish hull-first
The classic demonstration of Earth's curvature is a ship sailing away to sea. As it recedes, it does not simply shrink to a dot; instead, its hull disappears first, sinking below the horizon, while the mast and sails remain visible above. Eventually even the top of the mast slips out of sight. A ship approaching from afar appears in the reverse order, the tip of its mast rising first, then the rest of the vessel climbing into view as it nears.
This happens because the bulge of water between you and the ship grows taller, relative to your line of sight, the farther the ship travels. First it hides the waterline, then the deck, then the superstructure, working its way up the vessel. The same effect hides the base of a distant lighthouse, the foot of a far mountain, or the lower floors of a skyscraper across a bay. The calculator's hidden-height figure is exactly the height of the object swallowed in this way, which is why sailors and surveyors have long relied on such calculations.
The eight inches per mile rule
A popular rule of thumb says the Earth curves away by about eight inches over the first mile. It captures the idea that the drop grows with the square of the distance: roughly eight inches at one mile, but about 32 inches at two miles, and so on, climbing steeply rather than evenly. It is a handy mental shortcut for getting a rough sense of the curvature over short distances.
However, the simple rule is only an approximation, and it overstates the drop at long range while ignoring the crucial effect of the observer's height. The amount of a distant object actually hidden depends not just on its distance but on how high your eyes are, since a higher viewpoint pushes your horizon farther out and hides less. This calculator uses the full geometry rather than the rough rule, accounting for your eye height, so its hidden-height results stay accurate even over long distances where the eight-inches rule breaks down.
Units and precision
The calculator takes distances and heights in units from millimetres to nautical miles, and uses Earth's mean radius of about 6,371 kilometres. It reports both the horizon distance and the hidden height of the distant object. The calculation is exact geometry for a smooth spherical Earth and ignores atmospheric refraction, which in reality bends light slightly downward and lets you see a little farther, hiding a touch less than pure geometry suggests.
A worked example
Suppose your eyes are 2 metres above the sea, and you are looking at an object 20 kilometres away.
The horizon distance for a 2-metre eye height works out to about 5 kilometres, so the object at 20 kilometres lies well beyond it. The curvature hides about 17.5 metres of the object's base, meaning anything shorter than that would be completely out of sight, and a taller structure would show only the part above 17.5 metres. This is why, from a low vantage, the hull of a distant ship or the foot of a far shoreline disappears while the upper parts remain visible.
Questions people ask
How much does the Earth curve over a distance?
The surface drops by about 8 centimetres over 1 kilometre, growing with the square of the distance, so about 7.8 metres over 10 kilometres. How much of an object is hidden also depends on your eye height.
Why do ships disappear from the bottom up?
Because the curve of the sea hides the lower part first. As a ship sails away, its hull sinks below the horizon while the mast stays visible, which only a round Earth explains.
Is the eight inches per mile rule accurate?
Only roughly, and only over short distances. It overstates the drop at long range and ignores your eye height. The full geometry, which this calculator uses, is more accurate.
How far is the horizon at sea level?
About 4.7 kilometres for eyes 1.7 metres above the water, and about 5 kilometres from 2 metres up. A higher viewpoint pushes the horizon farther out.
References
A quick note on where the geometry comes from. The horizon distance and curvature drop follow from spherical geometry, standard results in surveying and navigation, described by San Diego State University's atmospheric optics resource. The curvature correction for sight lines is a standard surveying calculation. NASA provides Earth's radius.
- San Diego State University, Andrew Young, Distance to the Horizon. https://aty.sdsu.edu/explain/atmos_refr/horizon.html
- NASA, Earth Fact Sheet (mean radius). https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html
- NOAA National Geodetic Survey, geodetic leveling and curvature correction. https://geodesy.noaa.gov/
Bibek Lal Karna is a PhD student and graduate teaching assistant at the University of Mississippi, with deep interests in theoretical and gravitational physics. He is also the founder of NRCC and is strongly engaged in scientific teaching and communication. At Eon Tools, he reviews physics tools.