Earth Curvature Calculator — Horizon Distance, Curvature Drop & Hidden Height

Earth Curvature Calculator: Horizon Distance, Curvature Drop & Hidden Height

Have you ever watched a ship sail toward the horizon and noticed how it seems to sink into the sea, hull-first, before the mast disappears? That is the curvature of the Earth in action. Our Earth Curvature Calculator lets you quantify exactly how much of a distant object is hidden below the horizon, how far you can see before the surface curves away, and how much of a target remains visible at any given distance.

What Is Earth's Curvature?

Earth is not flat — it is an oblate spheroid with a mean radius of approximately 6,371 km (3,958.8 miles). Because the surface curves away from any observer, objects at a distance gradually drop below the line of sight. The further away an object is, the more of it disappears below the visual horizon.

The classic demonstration is watching a tall ship sailing away: the hull vanishes first while the mast remains visible. This happens because the hull is closer to the water surface and therefore drops below the horizon sooner than the elevated mast.

The "8 Inches Per Mile" Rule

A popular approximation states that the Earth curves approximately 8 inches per mile squared (about 20 cm per km squared). More precisely, the curvature drop d over a distance D is:

drop = D² / (2 × R)

where R is Earth's radius (6,371,000 m in metric, or 20,902,464 ft in imperial) and D is the distance in the same primary units (meters or feet). For example, over 1 mile (5,280 ft):

drop = 5280² / (2 × 20,902,464) ≈ 0.667 ft ≈ 8 inches

This approximation holds well for distances up to a few hundred kilometers, beyond which more precise spherical geometry is needed.

How Far Can You See? — The Horizon Formula

The distance to the geometric horizon depends on the height of the observer's eyes above the ground (or sea level). The formula is:

d_horizon = √(2 × R × h)

where h is the observer height in primary units (meters or feet) and the result is in the same primary units. The calculator then converts this to km or miles for display.

• A person standing at sea level (eye height ≈ 1.7 m) can see roughly 4.65 km to the horizon.
• From a 10-meter cliff, the horizon extends to about 11.3 km.
• From a 100-meter lighthouse, you can see approximately 35.7 km.
• From a commercial aircraft at 10,000 m, the horizon is over 357 km away.

Note: this is the geometric horizon. Atmospheric refraction can extend the visible horizon by roughly 7–15% under standard conditions.

Calculating the Hidden (Obstructed) Height

When you know how far away a target object is, you can calculate how much of it is hidden below the horizon. The process is:

1. Calculate the curvature drop at the target distance: drop = D² / (2R)
2. Subtract the observer's eye height to find the net hidden height: hidden = max(0, drop − observer_height)
3. If you know the total height of the target, find the visible portion: visible = max(0, target_height − hidden)

For example, if you are 1.7 m tall standing at sea level and looking at a 50 m tall building 20 km away:

• Curvature drop at 20 km: 20,000² / (2 × 6,371,000) ≈ 31.4 m
• Hidden height: 31.4 − 1.7 = 29.7 m
• Visible height: 50 − 29.7 = 20.3 m — just over 40% of the building remains visible

Metric vs. Imperial Units

The calculator supports both unit systems:

• Metric: observer height and target height in meters (m), distance in kilometers (km), Earth radius = 6,371,000 m
• Imperial: observer height and target height in feet (ft), distance in miles (mi), Earth radius = 20,902,464 ft (3,958.8 miles); 1 mile = 5,280 feet

All internal calculations are performed in the primary unit of the selected system (meters or feet) before converting distances back to km or miles for display.

Practical Applications

• Navigation: Sailors and pilots use curvature calculations to determine when landmarks or other vessels become visible over the horizon.
• Photography & Videography: Photographers planning seascape or landscape shots need to know exactly how much of a distant subject will be visible.
• Engineering: Engineers designing communication towers, radar installations, and long-distance optical links must account for line-of-sight obstructions caused by Earth's curvature.
• Astronomy & Surveying: Surveyors apply curvature and refraction corrections for precise long-distance measurements.
• Education: Demonstrating Earth's curvature is one of the most compelling proofs that the Earth is spherical — easily verified with binoculars and a distant ship.

Frequently Asked Questions

Does atmospheric refraction affect the results?

Yes. The calculator uses the geometric (vacuum) model, which does not account for atmospheric refraction. Under standard atmospheric conditions, refraction effectively reduces the apparent curvature by about 7%, meaning objects can appear about 7–15% further away before they drop below the horizon. For engineering applications, a standard refraction coefficient of k = 0.13 is commonly used, replacing R with an effective radius of R / (1 − k) ≈ 7,320 km.

Why is the observer height included in the formula?

Because your elevated vantage point already "sees over" part of the curvature. The higher your eyes are above the surface, the further your horizon extends and the less of a distant target is hidden. An observer at sea level (h = 0) has zero horizon distance and every object beyond 0 km is technically hidden; in practice you need h > 0 for any visibility.

Can I use this calculator to verify that the Earth is round?

Absolutely. If you watch a tall building or ship from sea level and then climb to a higher elevation, the portion of the target that becomes visible matches the predictions of the spherical-Earth model extremely well. The "8 inches per mile squared" approximation is accurate to within a few percent for distances under 100 km, and is easily verified with a camera and a telephoto lens aimed at a distant building.

What if my target height is 0?

If no target height is given, the calculator still shows the curvature drop and hidden height at the specified distance. The visibility status then simply indicates whether the ground surface at that distance is above or below the observer's horizon (i.e., whether curvature drop exceeds observer height).

Is 6,371 km the exact radius of the Earth?

No — Earth is slightly flattened at the poles. The polar radius is approximately 6,357 km and the equatorial radius is about 6,378 km. The value of 6,371 km used here is the mean radius, which gives accurate results for most practical purposes. For geodetic precision, latitude-specific calculations would be required.

Why does the calculator use feet for imperial heights but miles for distance?

Because those are the natural units in each context: heights of people and buildings are measured in feet, while longer distances are measured in miles. Internally the calculator converts miles to feet (1 mi = 5,280 ft) so all calculations are performed in a single consistent unit before converting results back for display.