The Earth curvature calculator above answers three related but distinct questions. How far can you see before the surface falls away? How far below a flat sightline does the surface drop over a given distance? And how much of a distant object does that curve actually conceal? Those three numbers are routinely treated as interchangeable and they are not. The last one is the one that matters when you are looking at something and wondering why only its upper half is showing.
Arb Digital builds free calculators that carry every correction the real problem needs, and here that means refraction. Air is denser near the ground, so light travelling roughly horizontally bends gently downward and follows the curve part of the way. The standard surveying allowance treats the Earth as though its radius were seven-sixths of its true value, which reduces the apparent drop by about fourteen per cent. Leaving it out is the usual mistake, and it produces answers that disagree visibly with what anyone can go outside and observe.
What This Earth Curvature Calculator Does
You give it your eye height, the distance to something, and how tall that something is. It returns the height of the object hidden below the horizon as the headline figure, and four supporting numbers: how far your own horizon is, how far the surface drops below a flat line leaving your eye over that distance, how much of the object is still visible, and how high you would need to get for its top to appear at all.
The last of those is the practically useful one. If a target is entirely hidden, the tool tells you the eye height at which its top edge would break the horizon, which is the number a surveyor, a sailor or a photographer actually needs. It is not a linear relation — doubling the required visibility distance quadruples the height needed — so guessing at it goes wrong quickly.
Refraction is a dropdown rather than a hidden constant because it genuinely varies. The standard seven-sixths figure is the surveying convention for ordinary daytime conditions. Radio and microwave work uses four-thirds, because longer wavelengths bend more. Over a cold sea under a warm air mass, refraction can be far stronger still and produce looming and mirages that no fixed coefficient describes. Setting refraction to none gives the pure geometric answer, which is useful as a baseline and wrong as a prediction.
How to Use It
- Measure eye height, not your height. Eye height above the surface is what sets the horizon distance. On a beach that is roughly 1.7 m for a standing adult; on a cliff it is the cliff height plus your eye height.
- Use surface distance, not line-of-sight distance. Over the ranges where curvature matters these differ negligibly, but the figure to enter is the distance along the ground, which is what a map or a GPS gives you.
- Enter the full height of the object. The tool subtracts the hidden portion from it. If you only know the height of a feature part-way up, the visible-part figure will be misleading.
- Leave refraction on standard unless you have a reason. Pure geometry is a teaching baseline. Real observations, including every one you can make yourself, include refraction.
- Compare the drop figure with the hidden height. The gap between them is the whole reason curvature arithmetic gets argued about, and seeing the two side by side settles it faster than any explanation.
The Formula: Horizon, Drop and Hidden Height
Take R as the Earth's mean radius, 6,371 km — a single sphere standing in for a shape that surveying handles with reference ellipsoids such as GRS 80, described on the NOAA National Geodetic Survey datums page — and multiply it by the refraction factor to get an effective radius R′. The distance to the horizon from eye height h is then dh = √(2R′h + h²), and since h is tiny compared with R′ the second term almost never matters. The drop of the surface below a flat tangent line over a distance d is √(d² + R′²) − R′, which for everyday distances is very close to d² ÷ 2R′.
Hidden height is the one that needs care. Work out your horizon distance dh first. If the object is nearer than that, nothing is hidden. If it is further, the excess distance is d2 = d − dh, and the hidden height is d2² ÷ 2R′. The distance is measured from the horizon, not from you, which is why hidden height is always smaller than the drop over the same total distance.
Work the defaults. With standard refraction, R′ = 6,371 × 7/6 = 7,432.8 km. From an eye height of 2 m the horizon is √(2 × 7,432,833 × 2) = 5,453 m, about 5.45 km. An object 40 km away is 34,547 m beyond that horizon, so the hidden height is 34,547² ÷ (2 × 7,432,833) = 80.3 m. A 30 m building is therefore entirely out of sight, and to see its top you would need an eye height of about 24 m. The drop below a flat sightline over the same 40 km is 107.6 m — a third larger than the hidden height, and not the same quantity.
Refraction Is Not Optional
Light in the atmosphere does not travel in straight lines. Density falls with altitude, the refractive index falls with it, and a ray travelling nearly horizontally curves gently downward, following the Earth's surface for part of the drop. The standard way to handle this is not to trace curved rays but to pretend the rays are straight and the Earth is bigger, which is where the seven-sixths effective radius comes from.
The size of the effect is easy to underestimate. Turning refraction off in the default scenario raises the hidden height from 80.3 m to 95.9 m and pulls the horizon in from 5.45 km to 5.05 km. On the drop figure specifically, the standard correction reduces the geometric value by about fourteen per cent, since the drop is inversely proportional to the effective radius. Over long sightlines that difference is tens of metres, which is more than enough to change whether a landmark is visible.
The effect is real and measured, not a convenience. NOAA's solar calculation details page assumes 0.833 degrees of atmospheric refraction for sunrise and sunset calculations, which is why the Sun is geometrically already below the horizon at the moment it appears to touch it. NOAA also notes that small atmospheric variations have a larger effect when the object is near the horizon, which is exactly the regime this calculator works in — so treat any refraction figure as a typical value rather than a constant.
Drop, Hidden Height and the Eight Inches Rule
The rule of thumb that curvature amounts to roughly eight inches per mile squared describes the drop below a tangent line, and only that. It is a decent approximation of d² ÷ 2R for the pure geometric case, in imperial units, from an observer at zero height. It says nothing about what is visible.
Applying it to visibility is where it goes wrong, and it goes wrong twice. It ignores the observer's eye height, which pushes the horizon out and removes the first part of the distance from the calculation entirely. And it ignores refraction, which reduces what remains by another fourteen per cent or so. A calculation that uses the rule directly to argue about what should or should not be visible across a lake is answering a different question from the one being asked.
The correct sequence is always the same: horizon distance from the observer's height first, remaining distance second, hidden height from that remainder third. This calculator does it in that order, which is why its hidden-height figure is smaller than the naive one and matches what you can verify yourself with a telephoto lens and a known landmark.
Where This Sits Next to Our Other Distance Tools
This page is about vertical concealment by a curved surface. The great circle distance calculator is about horizontal distance across that same curved surface, between two sets of coordinates — it answers how far apart, while this answers what is in the way. Feed its output into the distance field here and the two work as a pair.
For line-of-sight radio rather than optical visibility, the free space path loss calculator covers signal strength once a path exists, and the four-thirds refraction setting on this page is the standard assumption for establishing whether that path exists at all. The index of refraction calculator handles refraction at a defined boundary, which is a different problem from the continuous gradient in the atmosphere. For the geometry itself, the arc length calculator and the 3D distance calculator are the general-purpose tools, and the angle converter is useful when a dip angle is quoted in degrees or minutes of arc.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Omitting refraction — the standard correction reduces the drop by about fourteen per cent and pushes the horizon further out. Every real observation includes it.
- Confusing drop with hidden height — the drop is measured from a flat line at your eye, the hidden height from the horizon. The hidden height is always the smaller figure, and it is the one that decides visibility.
- Ignoring your own eye height — it sets the horizon distance and removes the first stretch of the sightline from the calculation entirely. Assuming an observer at zero height overstates concealment substantially.
- Using the eight-inches-per-mile-squared rule for visibility — it approximates the tangent drop for a zero-height observer with no atmosphere. It answers a different question.
- Assuming a fixed refraction coefficient — it varies with temperature gradient, and over water with a strong inversion it can be several times the standard value, which is what produces looming and superior mirages.
Related Free Tools From Arb Digital
Pair this with the great circle distance calculator to get the surface distance between two coordinates before you check what is hidden. For radio line-of-sight work, the free space path loss calculator takes over once the path is established. The index of refraction calculator covers bending at a sharp boundary, and the arc length calculator, 3D distance calculator and angle converter handle the surrounding geometry. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
From an eye height of 1.7 metres with standard refraction the horizon is about 5 kilometres away. The distance goes as the square root of height, so doubling your eye height moves the horizon out by about 41 per cent, not by double.
Drop is how far the surface falls below a flat line leaving your eye, measured over the whole distance. Hidden height is how much of a distant object sits below the horizon, measured only over the distance beyond your horizon. Hidden height is always the smaller number and it is the one that determines what you can see.
The standard surveying allowance treats the Earth as having a radius seven-sixths of its true value, which reduces the curvature drop by roughly fourteen per cent and pushes the horizon further away. Omitting refraction is the most common error in curvature calculations and produces answers that do not match observation.
It approximates the drop below a tangent line for a zero-height observer with no atmosphere. It is not a visibility rule. It ignores the observer's eye height and refraction, both of which reduce how much of a distant object is actually concealed.
Because it depends on how quickly air density falls with height, which depends on the temperature gradient. Over a cold sea beneath warm air the gradient is much steeper than standard, refraction is much stronger, and distant objects can appear raised or stretched. That is why the calculator offers several settings rather than one constant.
The mean radius of 6,371 kilometres, multiplied by the refraction factor you select. The Earth is an oblate spheroid, so the true radius varies by about 21 kilometres between equator and pole, which is a third of a per cent and well below the uncertainty introduced by refraction.
High enough that your horizon reaches to within the object's own horizon distance. The calculator reports that eye height directly. Because horizon distance goes as the square root of height, the required height rises as the square of the distance you are trying to cover.
This tool is provided for educational and study use. It models a smooth spherical Earth with a single refraction coefficient and does not account for terrain, tides, temperature inversions or the varying atmospheric conditions that affect real sightlines. It is not a substitute for a survey.
