Earth Curvature Calculator — Horizon Drop & Hidden Height
Find out how much Earth's surface drops below a horizontal line of sight over any distance. Enter the distance to the target, your observer height, and whether to account for standard atmospheric refraction to get the curvature drop, hidden height, and your personal horizon distance.
km
m
Atmospheric refraction
Vertical drop of Earth's surface at that distance, adjusted for standard atmospheric refraction (k = 0.13)
- 1
Distance in metres
10 × 1000 = 10,000 - 2
Angle θ = d ÷ R
10,000 ÷ 6,371,000 = 0.00156961 - 3
Geometric drop = R × (1 − cos θ)
6,371,000 × (1 − cos(0.00157)) = 7.8481 - 4
Effective drop with refraction k = 0.13
7.8481 × (1 − 0.13) = 6.828Standard atmospheric refraction reduces the apparent drop by about 13%.
How does this calculator work?
Earth's surface curves away from a flat line by R × (1 − cos(d/R)) metres, where R = 6 371 km and d is the distance. At 10 km the drop is ~7.9 m; at 50 km it's ~196 m. Standard atmospheric refraction (k = 0.13) reduces the apparent drop by ~13%. The horizon from observer height h is √(2Rh/(1−k)) km away.
Formula
How this is calculated
Earth is an oblate spheroid with a mean radius of approximately 6 371 km (WGS-84). Over any horizontal distance d, the planet's surface curves away from a flat line of sight. The exact geometric drop is R × (1 − cos(d/R)) metres, which simplifies to d²/(2R) for short distances (within 0.1% for d < 100 km). For a 10 km distance, the drop is about 7.85 m; for 50 km, about 196 m.
Atmospheric refraction bends light slightly downward along Earth's surface, making distant objects appear higher than pure geometry predicts. Under standard atmospheric conditions (sea-level temperature ~15 °C, lapse rate 6.5 K/km), the refraction coefficient k ≈ 0.13. This effectively replaces Earth's geometric radius with a larger apparent radius of R/(1 − k), reducing the apparent drop by about 13%.
The 'hidden height' is how much of a target object is concealed below the horizon from an observer at a given eye height. An observer at 1.7 m (average adult eye level) has a geometric horizon of about 4.7 km; with standard refraction, that extends to roughly 4.9 km. The calculator plots drop vs distance so you can see how curvature accumulates non-linearly. Note that actual visibility is also affected by atmospheric haze, temperature inversions, and the target's contrast.
Frequently asked questions
The drop is purely vertical — it measures how far below a perfectly flat horizontal line the ground surface lies. At 10 km the drop is less than 8 m, which is invisible at ground level because mountains, buildings, and atmospheric haze are far larger effects. At 50 km the drop reaches about 170–200 m (with refraction), which starts to be significant for line-of-sight applications like radio and laser links.
Air density decreases with altitude, bending light rays downward in a gentle arc that follows Earth's curvature partially. The standard refraction coefficient k ≈ 0.13 reduces the apparent drop by about 13% and extends visible horizon distance by roughly 7%. Under temperature inversions or over warm surfaces, refraction can be much stronger (super-refraction), occasionally letting you see objects far beyond the normal horizon.
Yes — this calculator uses standard spherical-Earth geometry (mean radius 6 371 km). Over distances where the curvature drop exceeds the height of a visible object, that object should be partially hidden. Photographers and surveyors routinely verify this: a ship's hull disappears over the horizon before its mast, consistent with a spherical Earth. Atmospheric refraction must be accounted for because it partially offsets the geometric drop.
TG we-Calculate Editorial Team. (2026). Earth Curvature Calculator — Horizon Drop & Hidden Height [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/earth-curvature-calculator
TG we-Calculate Editorial Team. "Earth Curvature Calculator — Horizon Drop & Hidden Height." TG we-Calculate. 2026. https://we-calculate.com/calculator/earth-curvature-calculator.
TG we-Calculate Editorial Team, "Earth Curvature Calculator — Horizon Drop & Hidden Height," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/earth-curvature-calculator
@misc{wecalculate_earth_curvature_calculator, title = {Earth Curvature Calculator — Horizon Drop & Hidden Height}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/earth-curvature-calculator}}, year = {2026}, note = {TG we-Calculate} }
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