Earth Curve Calculator: How It Works

This Earth curve calculator works out how much of a distant object is hidden by the curvature of the Earth, given your eye height, the object’s height, and the distance to it. You enter those three values, and it returns the hidden amount, the visible height, and the horizon distance. The math uses the standard spherical-Earth model with a mean radius of 6,371 km.

Formula

The hidden height is calculated using the standard line-of-sight drop formula for a spherical Earth:

Hidden height = (d² / (2R)) – √(2Rh) + h

Where:

• d = distance to the object (in meters, converted from km)
• R = Earth’s mean radius = 6,371,000 m
• h = eye height above the ground (in meters)

The first term, d²/(2R), is the drop of the Earth’s surface over distance d. The second term, √(2Rh), is the distance to the horizon from your eye. If the object is beyond the horizon, the hidden amount is the drop minus the horizon distance plus your eye height. If the object is closer than the horizon, the hidden height becomes negative, meaning nothing is hidden.

Worked Example

Using the default values: eye height = 1.7 m, object height = 10 m, distance = 20 km.

1. Convert distance to meters: 20 km = 20,000 m.
2. Calculate the drop: (20,000)² / (2 × 6,371,000) = 400,000,000 / 12,742,000 ≈ 31.39 m.
3. Calculate the horizon distance from your eye: √(2 × 6,371,000 × 1.7) = √(21,661,400) ≈ 4,654 m (4.65 km).
4. Hidden height = 31.39 – 4,654 + 1.7 ≈ 31.39 – 4.65 + 1.7 = 28.44 m.

The calculator rounds to 28.44 m hidden. Since the object is 10 m tall, the visible height is 10 – 28.44 = -18.44 m, meaning the entire object is hidden. The horizon is 4.65 km away.

How to Use the Calculator

Enter your eye height (typically 1.5–2 m for a standing person, or higher if on a boat or building), the object’s height (e.g., a lighthouse, building, or mountain), and the distance to it. The calculator then shows:

• Hidden height – how much of the object is blocked by the curve.
• Visible height – how much remains above the horizon (negative means fully hidden).
• Horizon distance – how far you can see before the surface drops away.

If the hidden height is less than the object’s height, the top of the object is visible. If it’s greater, the object is completely below the horizon.

Common Mistakes and Misconceptions

A frequent error is ignoring refraction. The atmosphere bends light slightly, making the horizon appear farther and reducing the hidden height. A common rule of thumb is to multiply the hidden height by about 0.83 for standard refraction, but this varies with temperature and pressure. The calculator here assumes no refraction, so it gives a conservative (slightly larger) hidden height.

Another mistake is mixing units. The formula requires meters for all distances. If you enter distance in miles or feet, the result will be wildly wrong. Convert first: 1 mile = 1,609 m, 1 foot = 0.3048 m.

People also confuse the drop of the Earth (d²/(2R)) with the hidden height. The drop is measured from a tangent line at your location, but your eye is above the ground, so the horizon extends farther. The hidden height is smaller than the drop by roughly the horizon distance.

When to Use This Calculator

This calculator is useful for:

• Photographers planning shots of distant landmarks or ships.
• Boating and sailing – estimating when a lighthouse or coastline will appear.
• Amateur radio – predicting line-of-sight over water.
• General curiosity about how far you can see from a hill or building.

It works best over water or flat terrain where the ground doesn’t rise between you and the object. Over land, hills and buildings can block the view sooner than the Earth’s curve.

Limitations of the Model

The calculator assumes a perfectly spherical Earth with a mean radius of 6,371 km. In reality, the Earth is an oblate spheroid – slightly wider at the equator. The difference is small (about 0.3%), so it has negligible effect on the result. More importantly, the model ignores:

• Atmospheric refraction (bends light, making the horizon appear farther).
• Terrain elevation between you and the object.
• Wave height and tidal variations over water.

For extreme distances (hundreds of km), the curvature formula becomes less accurate because the Earth’s radius varies slightly, and refraction plays a larger role. For most practical distances (under 50 km), the error is less than a few percent.

Frequently Asked Questions

Does this calculator account for refraction?

No, it assumes a straight line of sight. In reality, light bends downward due to atmospheric density gradients, which makes the hidden height smaller. A typical correction is to multiply the hidden height by 0.83, but this is an approximation. For precise results, you’d need temperature and pressure data.

Why does the hidden height come out negative?

A negative hidden height means the object is closer than your horizon, so no part of it is hidden. The calculator shows a negative number, but you can interpret it as zero hidden – the entire object is visible.

Can I use this for a drone or a tall building?

Yes, just enter the height of the drone or building as the object height, and your eye height as 0 if you’re at ground level. For a drone at 120 m and an eye height of 1.7 m, the horizon is about 40 km away, so the curve only matters beyond that.

What is the difference between drop and hidden height?

The drop is how much the Earth’s surface falls away from a tangent line at your location. The hidden height is the drop minus the distance to your horizon (plus your eye height). The horizon distance is about 4.7 km for a 1.7 m eye height, so the hidden height is always less than the drop.

How far can I see from a typical beach?

From eye height 1.7 m, the horizon is about 4.65 km away. A 10 m tall boat would be visible until its bottom dips below the horizon – roughly 11.3 km away (horizon distance + distance to the boat’s base). The calculator gives the exact number.

Is the Earth’s curvature noticeable in everyday life?

Not at short distances. Over 1 km, the drop is only about 8 cm, which is imperceptible. Over 10 km, the drop is about 7.8 m, which is noticeable when looking across a lake. The calculator is most useful for distances over 5 km.

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in h | Eye height | m | 1.7 | 0
in H | Object height | m | 10 | 0
in d | Distance to object | km | 20 | 0
out hidden | Hidden height | m | (d * 1000)^2 / (2 * 6371000) - sqrt(2 * 6371000 * h) + h | 2
out visible | Visible height | m | H - hidden | 2
out horizon | Horizon distance | km | sqrt(2 * 6371000 * h) / 1000 | 2