Earth Curvature Calculator (Horizon Distance & Hidden Height)
How far is the horizon, and how much of a distant ship, island or building is hidden by the curve of the Earth? Enter your eye height and the distance — with or without atmospheric refraction.
How it works
Horizon distance d₀ = √(2Rh + h²) · hidden height = √((D − d₀)² + R²) − R for D > d₀ · drop below a level line = R ÷ cos(D/R) − R ≈ D² ÷ 2R · R = 6,371 km (× 7/6 with standard refraction)
Because the Earth is a sphere with a mean radius of 6,371 km, the sea surface falls away from any level line of sight. Your eye height sets how far away the horizon is; anything beyond it loses its base below the horizon line, by an amount that grows roughly with the square of the extra distance. Light near the ground usually bends slightly downward with the curve, which is modelled by using an Earth radius 7/6 times larger — it makes the horizon about 8% farther and the hidden height about 14% smaller. On calm days with strong temperature layers, refraction can be much larger (mirages, "looming").
Worked example
Standing with your eyes 2 m above the sea, the horizon is √(2 × 6,371,000 × 2) = 5.05 km away. A boat 10 km away is 4.95 km beyond the horizon, so the bottom 1.92 m of it is hidden — of a 10 m tall boat you see the top 8.08 m. The sea surface at 10 km lies 7.85 m below a perfectly level line.
Sources: Wikipedia — Horizon (distance to the horizon) · NOAA — The Earth’s mean radius and geodesy basics
Frequently asked questions
How far away is the horizon?
About 3.57 × √(eye height in m) km without refraction. At 1.7 m eye height it is about 4.7 km (2.9 miles); from a 100 m cliff about 36 km.
Why is "8 inches per mile squared" wrong for hidden height?
That rule gives the drop below a level line from the point where you stand, not from your eye height. It ignores the observer’s height and the horizon distance, so it overstates how much is hidden.
What does refraction change?
The atmosphere bends light along the curve, so you see a little farther and less of distant objects is hidden. The standard correction multiplies the Earth’s radius by 7/6.
Can I see the curvature of the Earth?
From a plane at about 10–12 km the horizon dips about 3° and is roughly 360 km away, and a slight curve becomes noticeable with a wide view; from the ground it is too small to see directly.
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This calculator is for educational purposes. Double-check important results before acting on them.