Use this earth curvature calculator to estimate curvature drop, horizon distance and hidden height for an object at a given distance.
How to use the Earth curvature calculator
- Enter distance to object: In the “Distance to object” field, type how far away the target is along the surface, such as the shoreline, a building, or a ship.
- Pick distance units: Use the drop-down menu next to the distance box to choose km, m, mi, or ft so the calculator can convert the distance correctly.
- Open advanced options (optional): Click “Advanced options” if you want to set your eye height, change the planet radius, or include an atmospheric refraction factor.
- Set eye height (optional): In “Eyesight level above surface”, enter how high your eyes are above the ground or water and choose m or ft. Leave this blank to use a default eye height of 2 m.
- Adjust planet radius (optional): In the “Planet radius” row, type a different radius and unit if you want to model a body other than Earth. Leave this empty to keep the standard Earth radius.
- Include refraction coefficient (optional): In “Refraction coefficient k”, enter a value such as 0.13 if you want to account for typical atmospheric bending, or leave it blank to ignore refraction.
- Calculate and review results: Press “Calculate” to see the curvature drop at that distance, the horizon distance for your eye height, and how much of an object at that range would be hidden below the horizon based on your settings.
Earth curvature and the size of the planet
The curvature of the Earth comes from its roughly spherical shape. A simple average radius often used in calculations is about 6,371 kilometres. This value is an approximation because the real Earth is slightly flattened at the poles and bulged at the equator.
For many short distance problems, it is accurate enough to treat the Earth as a perfect sphere. This makes the geometry easier and allows simple formulas for drop, horizon distance and hidden height. More detailed models use an oblate spheroid or full geoid data, but these are rarely needed for basic line of sight questions.
When a calculator asks for planet radius, using 6,371 kilometres reproduces standard Earth curvature figures. Changing that radius lets you explore how horizons and hidden heights would look on other spherical worlds, such as a larger gas giant or a smaller rocky body.
Line of sight and the distance to the horizon
The distance to the horizon is the maximum range at which the surface is still visible from a given eye height. At sea level with eyes about 2 metres above the surface, the geometric horizon is a few kilometres away. As eye height increases, the visible distance grows because the line of sight can extend further before it grazes the curved surface.
In a simple spherical model, the horizon distance depends on the square root of the observer height times the planet radius. This is why climbing a hill, tower or building quickly increases how far you can see. For very high platforms such as aircraft, the horizon distance can reach hundreds of kilometres.
Earth curvature calculators use these relationships to show horizons for different observer heights. They can also show how the horizon changes if you adjust the radius to match another planet or moon.
Curvature drop and hidden height over distance
Curvature drop describes how far below a straight line the surface lies at some distance along the Earth. One common way to visualise this is to imagine a straight reference line drawn from the observer, then measure how far the curved surface drops away beneath that line. The drop grows faster than linearly with distance because of the spherical geometry.
Hidden height is a related idea. It asks how much of a distant object is blocked below the horizon when viewed from a given eye height. For example, a ship sailing away on the ocean will first lose its hull from view, then the superstructure, and finally the mast. An Earth curvature calculator can estimate the number of metres of an object that sit below the geometric horizon.
More precise models treat the Earth and objects as points on a circle and compute exact line of sight tangents. These methods account for both observer height and target height to decide whether the direct line between them clears the curved surface or intersects it.
Atmospheric refraction and apparent curvature
Light passing through the atmosphere does not travel in a perfect straight line. Layers of air with different temperature and density slightly bend light rays, an effect known as atmospheric refraction. Near the surface, standard conditions often bend light downward, which effectively lets observers see a bit farther than the pure geometric horizon would allow.
To handle this, some curvature calculations use an effective Earth radius that is slightly larger than the true physical radius. A refraction coefficient, often called k, describes how much the radius is scaled. A typical standard value increases the effective radius by a modest amount, which shifts the predicted horizon and hidden heights.
In strong temperature inversions or other unusual conditions, refraction can be much stronger than average. These situations can produce mirages or make distant objects appear higher or lower than simple geometric models predict.
Practical uses of Earth curvature calculations
Curvature and horizon calculations are used in many applied fields. Surveyors and geodesists account for curvature when they work over long baselines, so that level lines and height measurements remain consistent. Radio engineers consider both Earth curvature and refraction when planning microwave, radar or line of sight communication links.
In navigation, horizon distance helps estimate how far away features like lighthouses or islands can be seen from a given mast height. Map makers and software developers also use Earth models when converting between coordinates, distances and bearings on the curved surface.
An Earth curvature calculator gives quick numerical answers for drop, horizon distance and hidden height at different ranges, eye heights and refraction settings. This makes it easier to test scenarios and understand how the round shape of a planet affects visibility over distance.
FAQs
What does this Earth curvature calculator do?
This Earth curvature calculator estimates how much the surface curves away over a given distance. It shows the curvature drop below a straight line, the distance to the horizon from your eye height, and how much of a distant object would be hidden below the horizon. You can adjust eye height, planet radius, and atmospheric refraction.
What is the distance input used for?
The distance input is the straight-line distance along the surface from the observer to the object. The calculator converts this value into metres and uses it to compute curvature drop and the hidden part of an object at that distance.
What do the units for distance mean?
You can enter the distance in kilometres, metres, miles, or feet. The calculator converts all of these to metres internally, so the final curvature results are consistent regardless of the unit you choose.
What does curvature drop mean?
Curvature drop is the vertical difference between the surface at the observer and the surface at the target point if you draw a straight line from the observer that is tangent to the planet. It is the amount that the surface at the far end falls below that straight line because of the planet’s curvature.
What is the “Eyesight level above surface” option for?
The eyesight level input sets the height of the observer’s eyes above the surface. This affects the distance to the horizon and how much of a distant object is hidden. A higher eye height increases the horizon distance and reduces the part of the object that is obscured.
Why do I need to enter the planet radius?
The planet radius controls how strongly the surface curves. A larger radius means a gentler curvature and a smaller drop over the same distance. By default the calculator uses the mean radius of Earth, but you can change it to model other planets or to explore how sensitive the results are to radius.
What units can I use for planet radius?
You can enter the radius in kilometres or miles. The calculator converts the radius to metres before using it in the geometry formulas. The results are then converted back into user-friendly units for display.
What does the refraction coefficient k represent?
The refraction coefficient k approximates the effect of atmospheric refraction on line of sight. In standard conditions near Earth’s surface, k is often taken to be around 0.13. A positive k value effectively increases the Earth’s radius for visibility calculations, which increases the horizon distance and reduces the hidden height of distant objects.
How does refraction change the results?
When you enter a non-zero refraction coefficient, the calculator replaces the physical planet radius with an effective radius that is larger. This mimics the way light bends slightly downward in the atmosphere. The curvature drop relative to the physical radius stays the same, but the horizon distance and the hidden part of the object are reduced because the effective surface appears less curved to the observer.
What is the “obscured object part” value?
The obscured object part is the height of the lower section of an object that lies below the horizon line for a given distance, eye height, planet radius, and refraction setting. If this value is greater than zero, that part of the object would be hidden from view. If the calculator reports that the base is still above the horizon, the object would be fully visible in this simple model.
Does the calculator account for terrain, buildings, or waves?
No, the calculator assumes a smooth spherical planet and does not model local terrain, buildings, vegetation, waves, or other obstacles. It also assumes that the observer and the object are both located on the surface at the specified heights, with no elevation changes between them.
Is this Earth curvature model exact?
The calculator uses a simple spherical geometry model, which is accurate enough for most educational and practical visibility checks. It does not include full atmospheric ray tracing, Earth’s slight flattening at the poles, or detailed weather conditions. For high-precision survey work, professional geodetic models and measurements are needed.
Calculator Methodology and Sources
This Earth curvature calculator models the Earth or another planet as a perfect sphere with a chosen radius. It takes the surface distance from the observer to the target, converts it into metres, and uses basic circle geometry to compute the geometric drop due to curvature, the distance to the horizon from the eye height, and how much of a distant object would be hidden below the horizon.
The default planet radius is set to the commonly used mean radius of the Earth, about 6,371 kilometres (6,371,000 metres). This value is consistent with global geodetic and astronomy references that quote an average Earth radius of about 6,371 kilometres, such as NASA’s planetary size tables and geophysics summaries of Earth radius. See for example NASA Solar System sizes and the overview in NASA Earth Fact Sheet.
For a given surface distance along the planet, the calculator converts that distance to metres and treats it as an arc length on a circle of radius R. It then computes the curvature drop as the sagitta of that arc, using a standard circle relationship equivalent to drop = R × (1 − cos(d/R)), where d is the arc distance in metres. This gives the height difference between the local horizontal at the observer and the spherical surface at the target point. The approach follows the same right triangle geometry that appears in distance-to-horizon derivations in many treatments of the geometric horizon.
To estimate the distance to the horizon for a chosen eye height, the calculator uses the spherical geometry of a tangent line from the eye point to the sphere. It solves for the angle at the planet centre where the line of sight just grazes the surface, then multiplies that angle by the planet radius to obtain the surface distance to the geometric horizon. This is consistent with standard derivations of the horizon distance formula, which relate eye height, planet radius and horizon distance using Pythagoras on the right triangle formed by the radius, the radius plus eye height, and the line-of-sight tangent. For reference, see the discussion and formulas in NASA: Distance to the Horizon and educational notes such as the “How far away is the horizon?” handout from the University of Washington.
The “obscured object part” result uses a numerical search to find the minimum object height that allows a straight line from the observer’s eye to the object to just clear the spherical surface. The code increases the object height until the shortest distance between the line of sight and the planet centre equals the planet radius, which is the condition for the line just touching the sphere. Any extra height above this threshold is reported as the hidden portion below the geometric horizon. This method is a computational way to apply the same tangent-line geometry used for the horizon, but now treating the object height as the unknown.
The optional refraction coefficient k lets the calculator account approximately for bending of light in the atmosphere. It uses the common “effective Earth radius” method, where the geometric calculations are repeated with an adjusted radius R_eff = R / (1 − k). A typical value of k = 0.13 is often quoted in surveying and geodesy as a standard near-surface refraction coefficient under average conditions, for example in Gaussian refraction discussions and practical surveying references. See Hirt et al. on refraction modelling, and summaries such as “Optical Refraction Curvature in the Troposphere” that describe the origin and uncertainty range of the k ≈ 0.13 value.
These calculations assume a perfectly spherical planet with a single radius, a uniform refraction coefficient along the line of sight when refraction is enabled, and no local topography between observer and target. Real Earth conditions include mountains, buildings, sea waves, variable air density and temperature gradients, so actual visibility can differ from these idealised geometric and “standard refraction” results. The radius and k inputs are editable so that users can adjust them for different planets or for sensitivity checks around the standard Earth and atmosphere values.