Calculate the curvature drop, bulge, and hidden height of objects over distance on a spherical Earth with optional refraction correction.
The earth curvature calculator computes how much the surface curves away from a horizontal reference line over a given distance. On a spherical Earth with radius 6,371 km, the surface drops by approximately 7.85 cm per kilometer at close range, following a quadratic relationship that grows rapidly with distance.
This calculator is indispensable for surveyors, civil engineers, and telecommunications professionals who must account for curvature effects in leveling, bridge construction, long-range photography, and line-of-sight radio links. Over a distance of just 10 km, the curvature drop exceeds 7.8 meters — enough to completely hide a two-story building from a ground-level observer.
The tool also calculates the midpoint bulge (the maximum height of the Earth's surface between two points), hidden height of distant objects, and whether a target of known height is visible from the observer's position. An optional atmospheric refraction coefficient allows more realistic modeling of optical line-of-sight under various atmospheric conditions.
Understanding Earth's curvature is crucial for long-distance engineering projects, surveying, telecommunications link budgets, and even settling common misconceptions about visibility. Whether you're designing a bridge, setting up a microwave relay, or calculating whether a distant cityscape should be visible across a lake, this calculator provides instant, accurate answers with atmospheric refraction support.
Curvature drop: h = d² / (2·R) where d is horizontal distance and R is Earth's radius (6,371 km). Midpoint bulge: b = R − √(R² − (d/2)²). Hidden height: h_hidden = (d − d_horizon)² / (2R) where d_horizon = √(2·R·h_observer). Refraction uses effective radius R_eff = k·R.
Result: 7.85 m curvature drop at 10 km
At 10 km, the drop is (10000)² / (2 × 6,371,000) = 7.85 m. A person standing at ground level would not see any object shorter than 7.85 m at that distance.
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Approximately 8 inches (20 cm) for the first mile, but the drop grows quadratically — at 2 miles it is about 2.7 feet, and at 10 miles it is about 66.7 feet.
Drop is measured from a horizontal tangent line at the observer; bulge is the maximum height of the curved surface between two endpoints at the same elevation. Use this as a practical reminder before finalizing the result.
Yes. Standard atmospheric refraction effectively increases the Earth's radius by about 7%, bending light to follow the curvature slightly, which extends visibility.
No. The Earth is an oblate spheroid, slightly flattened at the poles. The equatorial radius is about 21 km larger than the polar radius. For most practical calculations, a sphere of R = 6371 km is sufficiently accurate.
From sea level, curvature is difficult to perceive directly. From high altitudes (10+ km) or in long-exposure photographs across large bodies of water, curvature becomes measurable.
Surveyors apply curvature and refraction corrections to leveling observations. The combined correction is approximately 0.0675 × d² meters where d is in kilometers.