Flat vs Round Earth Calculator — Horizon & Curvature Drop
Enter your eye height and a distance to find how far the horizon is on a round Earth and how many metres the ground curves away — predictions that differ from a flat-Earth model by measurable amounts even over short distances.
m
km
Distance at which the surface curves out of sight from your eye height
- 1
Convert observer height to km
h = 2 m ÷ 1,000 = 0.002 - 2
Compute radicand (2Rh + h²)
2 × 6,371 × 0.002 + 0.002² = 25.484004h² is negligible at human-scale heights but exact here. - 3
Horizon distance
√25.484004 = 5.05
How does this calculator work?
On a spherical Earth (R = 6,371 km), the horizon from eye height h is √(2Rh) km away and the surface curves d²/(2R) below flat at distance d. At 2 m eye height: horizon ≈ 5 km; at 10 km distance: curvature drop ≈ 7.85 m. A flat Earth predicts zero drop — directly contradicted by measurement.
Formula
How this is calculated
A spherical Earth curves away from any observer at a rate set by the Earth's mean radius R = 6,371 km. The horizon distance is the point where the surface becomes tangent to the observer's line of sight: d_horizon = √(2Rh + h²), where h is the eye height above the surface in the same units as R. For everyday heights (h much smaller than R) the h² term is negligible and the formula simplifies to √(2Rh). At 2 m eye height the horizon is about 5.0 km away; from the top of a 100 m lighthouse it extends to about 35.7 km; from a cruising aircraft at 10 km altitude it reaches about 357 km.
The curvature drop describes how far the Earth's surface has curved below a horizontal plane at a given distance d: drop ≈ d²/(2R). Over 1 km the drop is about 7.8 cm; over 10 km it is about 7.85 m; over 100 km it reaches about 785 m. These amounts have been confirmed by laser-level surveys, canal-construction engineering, and long-baseline photography. On a perfectly flat Earth the drop would be exactly 0 at any distance — a directly contradicted prediction.
This model assumes a smooth spherical Earth with no atmospheric refraction. In practice, standard atmospheric refraction bends light slightly downward, extending the optical horizon by roughly 7–8% compared with the geometric value derived here. Neither the geometric nor the refraction-corrected horizon is consistent with a flat-Earth model.
Frequently asked questions
Using drop = d²/(2R), the surface drops about 7.8 cm below a flat plane over 1 km, 31 cm over 2 km, and 7.85 m over 10 km. The drop grows with the square of the distance, making it increasingly significant at longer ranges.
Atmospheric refraction bends light slightly downward, effectively making the Earth's optical radius about 7–8% larger than its geometric radius. Unusually strong temperature inversions cause super-refraction or "looming", occasionally allowing objects far beyond the geometric horizon to become visible.
The formula drop ≈ d²/(2R) is an approximation that assumes d is much smaller than R. For distances up to a few hundred kilometres the error is small (< 1%). For very large arcs a more precise spherical-geometry formula should be used, but the approximation is adequate for all everyday observations.
Also known as
TG we-Calculate Editorial Team. (2026). Flat vs Round Earth Calculator — Horizon & Curvature Drop [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/flat-vs-round-earth-calculator
TG we-Calculate Editorial Team. "Flat vs Round Earth Calculator — Horizon & Curvature Drop." TG we-Calculate. 2026. https://we-calculate.com/calculator/flat-vs-round-earth-calculator.
TG we-Calculate Editorial Team, "Flat vs Round Earth Calculator — Horizon & Curvature Drop," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/flat-vs-round-earth-calculator
@misc{wecalculate_flat_vs_round_earth_calculator, title = {Flat vs Round Earth Calculator — Horizon & Curvature Drop}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/flat-vs-round-earth-calculator}}, year = {2026}, note = {TG we-Calculate} }
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