Advanced Earth Curve Calculator: See the Horizon

This advanced earth curve calculator works out the distance to the horizon, the hidden height of a distant object, and the apparent drop due to the Earth’s curvature. You enter your eye height, the target distance, and optionally the target’s height, and it returns the horizon distance, the hidden amount, and the line-of-sight drop. It uses the standard geometric model of a spherical Earth with a radius of 6,371 km (3,959 miles), ignoring atmospheric refraction for a pure, textbook answer.

Formula

The calculator uses the Pythagorean theorem applied to a circle. For an observer at height h (in meters) above the Earth’s surface, the distance to the horizon D (in km) is:

D = √((R + h/1000)² − R²)

where R is the Earth’s radius (6,371 km). For the hidden height H (in meters) of a target at distance d (in km) beyond the horizon, the formula is:

H = (d − D)² × 1000 / (2R)

This is the standard “8 inches per mile squared” rule converted to metric: the drop is roughly 0.0785 × d² meters when d is in km. The total drop (the amount the Earth’s surface falls away over a distance d) is:

Drop = d² × 1000 / (2R)

All variables: h = eye height in meters, d = distance to target in km, t = target height in meters (used only for the “visible if t > hidden” check, not in the main outputs), R = Earth radius in km.

Worked Example

Using the default values: eye height 1.7 m, distance 10 km, target height 0 m.

1. Convert eye height to km: 1.7 / 1000 = 0.0017 km.
2. Compute (R + h)² − R²: (6371.0017)² − 6371² ≈ 21.66 km².
3. Take the square root: √21.66 ≈ 4.65 km. This is the horizon distance.
4. Since 10 km > 4.65 km, the target is beyond the horizon. Hidden height = (10 − 4.65)² × 1000 / (2 × 6371) ≈ 28.6 × 1000 / 12742 ≈ 2.25 m.
5. Total drop = 10² × 1000 / 12742 ≈ 7.85 m.

The calculator shows: horizon ≈ 4.65 km, hidden height ≈ 2.25 m, total drop ≈ 7.85 m. This means that from 1.7 m eye height, the surface of a 10 km distant lake appears to drop about 7.85 m, and an object at water level would be hidden by about 2.25 m.

When to Use This Calculator

This tool is for geometric line-of-sight problems: how far you can see from a hilltop, whether a distant building or ship is visible, or how much of a mountain is hidden behind the curve. It is also used in surveying, photography (to predict if a distant subject will be visible), and by anyone planning long-distance observations. The calculator assumes a perfectly spherical Earth with no obstructions, no atmospheric refraction, and no terrain elevation changes.

For practical use, remember that the horizon distance depends only on your eye height, not on the target. Doubling your eye height increases the horizon by about 41% (since √2 ≈ 1.41). The hidden height grows with the square of the distance beyond the horizon, so doubling that distance quadruples the hidden amount.

How to Read the Results

The three outputs answer different questions:

• Horizon distance – how far you can see before the Earth’s surface blocks the view. For a standing person (1.7 m eyes), it’s about 4.65 km. From a 10 m cliff, it’s about 11.3 km.
• Hidden height – how much of a target at water level is cut off by the curve. If the target’s actual height (the third input) is greater than this, the top should be visible; otherwise it’s fully hidden.
• Total drop – the vertical drop of the Earth’s surface over the full distance, as if you had a straight line from your eye to the target point. This is the “8 inches per mile squared” figure.

For example, with eye height 1.7 m and a target at 20 km, the horizon is still 4.65 km, but the hidden height jumps to about 18.5 m. A 20 m tall building would just barely peek above the horizon.

Common Mistakes and Misconceptions

The most frequent error is mixing units: using miles with the metric formula or forgetting to convert eye height to km. Another is confusing “hidden height” with “total drop.” The total drop is always larger because it measures the full curve over the entire distance, while the hidden height only counts the part beyond your horizon. For example, at 10 km, the total drop is 7.85 m, but the hidden height is only 2.25 m – a big difference.

A common misconception is that the “8 inches per mile squared” rule gives the hidden height. It actually gives the total drop. For a 10 km distance (6.2 miles), the rule gives about 8 × 6.2² = 307 inches ≈ 7.8 m, which matches the total drop, not the hidden height. To get the hidden height, you must subtract the horizon distance first.

Also, this calculator ignores atmospheric refraction, which bends light downward and makes the horizon appear farther (by about 8–10% on average). The pure geometric answer is what you get here; real-world visibility is often slightly better.

Limits of the Formula

The spherical Earth model is accurate for distances up to a few hundred kilometers, but it fails for very long distances (thousands of km) where the Earth’s ellipsoidal shape and gravitational variations matter. It also assumes no terrain elevation: if you’re on a mountain, you must add the mountain’s height to your eye height. The calculator does not account for obstacles like buildings or trees, nor for the target’s elevation – you need to add that to the target height input manually.

For extreme precision (e.g., geodetic surveying), you would use a more complex model that includes the Earth’s equatorial radius (6,378 km) and polar radius (6,357 km). The difference is less than 0.5% for most practical distances, so this calculator is fine for general use.

Alternatives and Extensions

If you need to include refraction, you can adjust the Earth’s radius to about 7,430 km (a “refracted radius” that simulates the average bending of light). This changes the horizon distance to about √(2 × 7430 × h/1000) km, which is roughly 9% farther than the geometric value. Some advanced calculators let you toggle this on.

For line-of-sight between two elevated points (e.g., two mountain peaks), you need to compute the horizon for each point and then check if the sum of the two horizon distances is greater than the distance between them. This calculator handles one observer and one target; for two observers, you would run it twice with each person’s eye height.

For a quick mental estimate, use these approximations: horizon distance in km ≈ 3.57 × √(eye height in meters). Total drop in meters ≈ 0.0785 × (distance in km)². Hidden height in meters ≈ 0.0785 × (distance beyond horizon in km)².

Frequently Asked Questions

Why does the hidden height differ from the total drop? The total drop measures the Earth’s curvature over the entire distance from you to the target. The hidden height only counts the part of the target that is beyond your horizon. Since you can see the near part of the curve, the hidden portion is smaller. For a target at 10 km with eye height 1.7 m, the drop is 7.85 m, but only 2.25 m is actually hidden because the first 4.65 km are within your line of sight.

Can I see a mountain from 100 km away? It depends on your eye height and the mountain’s height. With eye height 1.7 m, the horizon is 4.65 km, so the hidden height at 100 km is about (100 − 4.65)² × 1000 / 12742 ≈ 713 m. A mountain must be taller than 713 m to be visible (ignoring refraction). Many mountains are, so yes, you can see tall peaks from 100 km away if the air is clear.

Does this work for the moon or sun? No – those are celestial bodies, and the line-of-sight is essentially straight because they are extremely far away. The Earth’s curvature has no effect on seeing the moon; you see it when it’s above your local horizon, which is a different geometric problem.

What is the “8 inches per mile squared” rule? It’s an approximation of the total drop in inches for a distance in miles: drop ≈ 8 × (miles)². For 1 mile, drop ≈ 8 inches; for 2 miles, ≈ 32 inches. It’s a quick way to estimate the Earth’s curvature, but it gives the total drop, not the hidden height, and it’s only accurate for distances under about 100 miles.

Why doesn’t the calculator include refraction? Refraction varies with temperature, pressure, and humidity, so it can’t be a fixed factor. The calculator gives the pure geometric answer, which is the worst-case (most conservative) estimate. To include average refraction, you can use a larger effective Earth radius (about 7,430 km), which you can manually account for by adjusting the inputs – but this calculator keeps it simple and transparent.

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in h | Eye height | m | 1.7 | 0
in d | Distance to target | km | 10 | 0
in t | Target height | m | 0 | 0
out horizon | Horizon distance | km | sqrt((6371 + h/1000)^2 - 6371^2) | 2
out hidden | Hidden height | m | max(0, d - horizon) * (d - horizon) * 1000 / (2 * 6371) | 2
out drop | Total drop | m | d * d * 1000 / (2 * 6371) | 2