Earth Curvature Calculator - Horizon Distance and Drop

An earth curvature calculator turns observer eye height and the Earth's radius into three practical numbers: the straight-line distance from the observer to the horizon, the curvature drop below the line of sight at any chosen distance, and the height of a distant object that is hidden behind that drop. Use it when planning an observation from a hilltop, lighthouse, drone, or high building.

• • Lighthouse and coastal visibility: Estimate the geographic horizon from a known lighthouse height so you can decide whether a shoreline, ship, or island falls inside the visible arc.
• • Drone and rooftop photography: Find out how much of a distant skyline, mountain, or coast is hidden by curvature at the camera height you actually plan to fly or shoot from.
• • Long-distance sighting and surveying: Predict how much a known landmark is expected to drop below the horizon at a given range.
• • Flat-earth and horizon tests: Compare the calculator's predicted horizon distance and obscured height against a real photograph or observation to test whether the geometry holds up.

The math behind the calculator is a single right triangle plus the Pythagorean theorem: one cathetus is the Earth's radius, the hypotenuse is the radius plus the observer's eye height, and the remaining cathetus is the line-of-sight horizon distance.

Pair this tool with the binoculars range calculator to estimate how far away a distant landmark actually is, and with the tree height calculator for the geometry behind any height-from-distance measurement you do in the field.

The earth curvature calculator takes your eye height, target distance, Earth radius model, and atmospheric refraction factor, then solves two right triangles that share Earth's centre. The first triangle gives the line-of-sight horizon. The second extends to the target and returns the height that the curve has hidden from view.

• earthRadius: Effective radius of the Earth. Default 6,371.0088 km (3,958.76 mi), with optional equatorial and polar values.
• eyeHeight: Height of the observer's eyes above the local reference surface.
• targetDistance: Surface distance from the observer to the distant object whose obscured height you want to find.
• refractionFactor: Fraction of the geometric drop removed by standard atmospheric refraction. 0 is no refraction, 1 is full vacuum, and 0.16 approximates a standard day.
• horizonDistance: Straight-line distance from the observer to the apparent horizon.

The curvature drop rule of thumb quoted by surveyors is about 8 inches per mile squared, which is the same expression 0.5 * d^2 / r written in imperial units.

According to Wikipedia - Earth Radius, the mean Earth radius is 6,371.0088 km, with an equatorial radius of 6,378.137 km and a polar radius of 6,356.752 km, which are the three Earth radius models this calculator offers.

According to Wikipedia - Atmospheric Refraction, the common surveying approximation for Earth curvature drop is 8 inches per mile squared, which the calculator surfaces as a separate coefficient so you can sanity-check the precise result against the rule of thumb.

If you would like to see how the same right triangle behaves on a flat cross-section of the sphere, the circle geometry calculator handles chord lengths, arc lengths, and central angles for any radius you choose.

Four small ideas explain every output of the earth curvature calculator. None of them is optional: the geometry uses all four at once.

These four ideas are linked by the same right-triangle identity. If you want to see the Pythagorean theorem that powers the horizon distance, the Pythagorean theorem solver walks through the algebra with simple numbers.

Six quick steps take you from a chosen eye height to a usable horizon distance and obscured height.

1. 1 Pick a unit system: Metric (m and km) or imperial (ft and mi).
2. 2 Enter observer eye height: Standing adult default is 1.7 m or about 5 ft 7 in.
3. 3 Enter target distance: Set to 0 to skip the obscured-height calculation.
4. 4 Choose an Earth radius model: Mean works for everyday use; switch to equatorial or polar for high-precision cases.
5. 5 Set the refraction factor: Use 0.16 for a standard day, 0 for vacuum-like conditions.
6. 6 Read the results: Horizon distance is the primary output. Secondary rows show curvature drop, obscured object height, and the 8 inches per mile squared coefficient.

Standing at a 100 m lighthouse with a clear view of the sea, set unitSystem to metric, eyeHeight to 100, targetDistance to 40, radiusMode to mean, and refractionFactor to 0.16. The calculator returns a horizon distance of 35.70 km, a drop at 40 km of about 125.57 m, and an obscured target height of 1.45 m. Point the same lighthouse at a coastline 33 km away and the obscured height drops to about 0.57 m.

A purpose-built earth curvature calculator turns three textbook formulas into a usable answer in seconds.

• • Saves trigonometry by hand: Horizon, drop, and obscured height are small variations of the same Pythagorean identity, but doing the arithmetic in your head invites mistakes.
• • Switches units without re-deriving: Pick metric or imperial; the calculator handles the kilometres-per-mile conversion.
• • Includes atmospheric refraction: Move from a pure geometric horizon to the apparent horizon you see on a normal day, with the option to dial it back to zero for high-altitude cases.
• • Confirms the 8 inches per mile rule: A separate rule-of-thumb value lets you sanity-check the result against the surveyor's quick estimate.

If you would rather see the same right-triangle geometry in its general form, the right triangle calculator solves the hypotenuse and cathetus lengths for any triangle with a 90 degree corner.

Three real-world factors shape the answer, and two limitations tell you when to double-check the result.

• • The model assumes a smooth sphere. Local terrain (mountains, valleys, cliffs) can hide or expose a target long before pure curvature does, so any line-of-sight estimate should be checked against the actual ground profile.
• • The refraction factor is a single number for a standard atmosphere. Strong temperature inversions over the sea, mirage conditions, and unusual humidity can push the apparent horizon far beyond the calculator's estimate.

According to Omni Calculator Earth Curvature page, The Omni Earth curvature calculator uses a = sqrt((r + h)^2 - r^2) for the horizon distance and x = sqrt(a^2 - 2ad + d^2 + r^2) - r for the obscured object height, both of which this calculator implements with the addition of an atmospheric refraction factor.

The same refraction physics that bends light around the Earth's bulge also bends it through a prism, so if you want to see how Snell's law changes the apparent elevation of a distant object, the angle of refraction calculator uses the same index-of-refraction idea with user-controlled angles.