What is Flat vs. Round Earth?

Use this calculator to compute Flat vs. Round Earth values with step-by-step solutions.

How do I use this calculator?

Enter the required values; results and step-by-step derivation update automatically.

Where is Flat vs. Round Earth used?

Astronomy & Geophysics, engineering, research, and related scientific disciplines.

What formulas does this use?

All standard flat vs. round earth formulas are applied and shown step-by-step in the results.

Can I get step-by-step solutions?

Yes. The calculator shows a full step-by-step derivation when results are displayed.

How accurate are the results?

Results use standard floating-point arithmetic (~15 significant digits). Suitable for education and professional use.

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PHYSICSAstronomy & GeophysicsPhysics Calculator

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Earth Curvature — Horizon and Hidden Height

On a spherical Earth (R ≈ 6371 km), horizon distance d ≈ √(2Rh). Hidden height behind curvature: h_hidden ≈ d²/(2R). Flat Earth predicts different values; experiments (ships, lighthouses) confirm round Earth.

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Horizon distance d ≈ √(2Rh) for observer height h Hidden height ≈ d²/(2R) — curvature hides distant objects ~8 in. drop per mile (approximate) Ships disappearing hull-first confirm curvature

Key quantities

d ≈ √(2Rh)

Horizon

Key relation

d²/(2R)

Hidden h

Key relation

6371 km

Earth R

Key relation

~8 in.

Drop/ mile

Key relation

Ready to run the numbers?

Why: Curvature affects visibility, navigation, and satellite line-of-sight. Round Earth geometry predicts horizon and hidden height; flat Earth gives different results. Observational tests distinguish models.

How: Enter observer height and distance. The calculator computes horizon distance, height hidden behind curvature, and angular drop. Compare round vs flat Earth predictions for the same inputs.

Horizon distance d ≈ √(2Rh) for observer height hHidden height ≈ d²/(2R) — curvature hides distant objects

Sources:NASA Earth Fact SheetNOAA Geodesy

Run the calculator when you are ready.

Solve the Curvature EquationsCalculate horizon distance and hidden height

🚢 Ship Disappearing Over Horizon

Classic observation: ship hull disappears before mast due to Earth curvature

🏗️ Lighthouse Visibility Distance

Calculate maximum distance a lighthouse can be seen from sea level

✈️ Airplane Flight Path Curvature

Observe Earth curvature from commercial flight altitude

🔬 Bedford Level Experiment

Famous experiment proving Earth curvature using poles and telescope

🛰️ Satellite Orbit Observation

Calculate visibility of satellites from ground level

Input Parameters

Observer Height

Observer Height Unit

Distance to Target

Distance Unit

Target Height

Target Height Unit

Earth Model

Advanced Options

Refraction Correction

Atmospheric Conditions

Display Settings

Decimal Precision

Display Format

For educational and informational purposes only. Verify with a qualified professional.

🔬 Physics Facts

🌍

Earth radius ~6371 km; horizon d ≈ √(2Rh) for height h.

— NASA

📐

Hidden height behind curvature ≈ d²/(2R) for distance d.

— NOAA

📏

Curvature drop ~8 in. per mile (approximate rule).

— USGS

🚢

Ships disappearing hull-first is classic curvature evidence.

— NOAA

📋 Key Takeaways

• Earth's mean radius is ~6,371 km; curvature drops ~8 inches per mile (12 cm per km)
• Horizon distance from sea level is ~4.7 km; higher observers see farther due to geometry
• Ships disappear bottom-first over the horizon—a key observation matching round Earth predictions
• Atmospheric refraction can extend visible horizon by ~8–15% by bending light rays
• Round vs flat Earth models differ in hidden height predictions; observations consistently match the spherical model

What is Earth Curvature?

Earth curvature refers to the geometric drop of the Earth's surface due to its spherical shape. As you look across a flat surface (like the ocean), objects gradually disappear behind the horizon because the Earth curves away from your line of sight.

The Earth has a mean radius of approximately 6,371 kilometers (3,959 miles), which means that for every mile you travel, the Earth's surface drops by about 8 inches(or 12 cm per kilometer). This curvature is measurable and observable in numerous real-world scenarios.

Key Facts:

Earth's circumference: ~40,075 km (24,901 miles)
Curvature drop: ~8 inches per mile (12 cm per km)
Horizon distance from sea level: ~4.7 km (2.9 miles)
Visible curvature requires altitude: ~35,000 ft (10.5 km)

How to Calculate Earth Curvature

The calculation of Earth's curvature involves several geometric formulas:

1. Horizon Distance

\text{The} \text{distance} \text{to} \text{the} \text{horizon} \text{from} \text{an} \text{observer} \text{at} \text{height} h \text{is} \text{given} \text{by}:

d = \\sqrt{2Rh + h^2}

For small heights (h << R), this simplifies to: d ≈ √(2Rh)

2. Hidden Height

\text{The} \text{height} \text{of} \text{an} \text{object} \text{hidden} \text{behind} \text{the} \text{curvature} \text{at} \text{distance} d:

h_{hidden} = \\sqrt{R^2 + d^2} - R - h_{obs}

This calculates how much of a distant object is obscured by Earth's curvature.

3. Angular Drop

\text{The} \text{angular} \text{drop} \text{in} \text{radians}:

\\theta = \\frac{d}{R}

Convert to degrees: θ° = θ × (180/π)

4. Curvature Drop (8 inches per mile)

\text{Simple} \text{approximation} \text{for} \text{curvature} \text{drop}:

h = 8 \\text{ inches/mile} \\times d_{miles}

This is approximately 12 cm per kilometer.

When to Use Earth Curvature Calculations

Earth curvature calculations are essential in many fields:

🌊 Maritime Navigation

Calculating visibility of lighthouses, ships, and coastal features from sea level. Critical for safe navigation and determining when objects appear or disappear over the horizon.

✈️ Aviation

Understanding horizon distance from different altitudes. Pilots use these calculations for navigation, landing approaches, and understanding visibility limitations.

📡 Telecommunications

Planning radio and microwave links. Earth curvature affects line-of-sight communications, requiring towers at appropriate heights to overcome curvature.

🔬 Scientific Experiments

Proving Earth's shape through experiments like the Bedford Level experiment, ship disappearing observations, and lighthouse visibility studies.

Famous Experiments Proving Earth's Curvature

1. Ship Disappearing Over Horizon

When observing a ship sailing away, the hull disappears before the mast. This is because the bottom of the ship is hidden first by Earth's curvature. At 10 km distance, approximately 7.8 meters of height is hidden behind the curvature.

2. Bedford Level Experiment (1870)

A famous experiment where poles of equal height were placed along a 6-mile stretch of the Bedford Level canal. When viewed through a telescope, the middle pole appeared higher than the end poles, proving Earth's curvature.

3. Lighthouse Visibility

Lighthouses are built tall to maximize visibility over the curved Earth. A 50-meter lighthouse can be seen from approximately 25 km away at sea level, but only the top portion is visible at greater distances.

4. Eratosthenes' Measurement (240 BC)

The ancient Greek mathematician calculated Earth's circumference by measuring shadow angles at two different locations, proving Earth was round over 2,000 years ago.

Round Earth vs Flat Earth Predictions

The key difference between round and flat Earth models is the prediction of hidden height:

Observation	Round Earth Prediction	Flat Earth Prediction	Actual Observation
Ship disappearing	Hull hidden first, then mast	Entire ship shrinks uniformly	✓ Matches Round Earth
Horizon distance	~4.7 km from sea level	Infinite (no horizon)	✓ Matches Round Earth
Sunset timing	Varies by location	Same time everywhere	✓ Matches Round Earth
Stick shadows	Different angles at different latitudes	Same angle everywhere	✓ Matches Round Earth

Atmospheric Refraction Effects

Atmospheric refraction can make objects appear slightly higher than they actually are, effectively increasing the visible horizon distance by approximately 8-15%. This is because light rays bend as they pass through air layers of different densities.

The calculator includes refraction correction options:

None: Pure geometric calculation without refraction
Standard: Typical atmospheric refraction (~8% increase)
Advanced: Refraction adjusted for specific atmospheric conditions

❓ Frequently Asked Questions

How far can I see on a round Earth?

The horizon distance depends on observer height. From sea level (1.7 m eye height), the horizon is approximately 4.7 km (2.9 miles) away. Formula: d = √(2Rh), where R is Earth's radius (6,371 km) and h is observer height. Higher observers see farther.

Why do ships disappear bottom-first over the horizon?

On a round Earth, the hull disappears first because it's closer to the water surface where curvature is most pronounced. The mast remains visible longer. On a flat Earth, the entire ship would shrink uniformly, which doesn't match observations.

How much of an object is hidden by Earth's curvature?

Hidden height increases with distance. At 10 km: ~7.8 m hidden. At 50 km: ~196 m hidden. At 100 km: ~785 m hidden. Formula: h = d²/(2R), where d is distance and R is Earth's radius.

Does atmospheric refraction affect curvature calculations?

Yes, refraction bends light rays, making objects appear slightly higher. Standard refraction increases visible distance by ~8-15%. This is why objects can sometimes be seen slightly beyond the geometric horizon. The calculator includes refraction correction options.

What is angular drop and how is it measured?

Angular drop is the angle between the horizontal and a line tangent to Earth's surface at the observer. At 1 km distance, angular drop is ~0.0086° (0.52 arcminutes). At 10 km: ~0.086° (5.2 arcminutes). It increases quadratically with distance.

Can I prove Earth is round with simple observations?

Yes! Watch a ship disappear bottom-first, observe different sunset times at different locations, measure shadow angles at different latitudes (Eratosthenes' method), or observe the horizon always forming a circle around you. All observations match round Earth predictions.

What is the Bedford Level experiment and what does it prove?

The Bedford Level experiment (1870) was intended to prove flat Earth but actually confirmed round Earth. When done correctly, it shows that objects at the same height appear at different elevations due to curvature. The original experiment had measurement errors.

How accurate are these curvature calculations?

The geometric formulas are accurate for ideal conditions. Real-world observations may vary due to atmospheric refraction, mirages, temperature inversions, and observer height measurement errors. For precise measurements, account for local conditions and use appropriate refraction corrections.

📚 Official Data Sources

USGS Earth Science Data

Earth measurements and geodesy

Updated: 2026-01-15

NOAA Geodesy

National Geodetic Survey data

Updated: 2025-12-01

NASA Earth Observatory

Earth observation data

Updated: 2025-11-20

National Geographic

Earth science education

Updated: 2025-10-15

Scientific American

Earth curvature research

Updated: 2026-01-10

⚠️ Disclaimer

This calculator provides geometric calculations based on a spherical Earth model (radius 6,371 km). Results assume ideal conditions without atmospheric effects. Real-world observations may vary due to atmospheric refraction, mirages, temperature inversions, and measurement uncertainties. The calculator includes refraction correction options, but actual conditions may differ. For precise measurements, consult geodesy references and account for local atmospheric conditions. Earth is an oblate spheroid (slightly flattened at poles), but the spherical approximation is accurate for most practical purposes.

👈 START HERE

⬅️Jump in and explore the concept!

Earth Curvature — Horizon and Hidden Height

🚢 Ship Disappearing Over Horizon

🏗️ Lighthouse Visibility Distance

✈️ Airplane Flight Path Curvature

🔬 Bedford Level Experiment

🛰️ Satellite Orbit Observation

Input Parameters

Advanced Options

Display Settings

🔬 Physics Facts

📋 Key Takeaways

What is Earth Curvature?

Key Facts:

How to Calculate Earth Curvature

1. Horizon Distance

2. Hidden Height

3. Angular Drop

4. Curvature Drop (8 inches per mile)

When to Use Earth Curvature Calculations

🌊 Maritime Navigation

✈️ Aviation

📡 Telecommunications

🔬 Scientific Experiments

Famous Experiments Proving Earth's Curvature

1. Ship Disappearing Over Horizon

2. Bedford Level Experiment (1870)

3. Lighthouse Visibility

4. Eratosthenes' Measurement (240 BC)

Round Earth vs Flat Earth Predictions

Atmospheric Refraction Effects

❓ Frequently Asked Questions

How far can I see on a round Earth?

Why do ships disappear bottom-first over the horizon?

How much of an object is hidden by Earth's curvature?

Does atmospheric refraction affect curvature calculations?

What is angular drop and how is it measured?

Can I prove Earth is round with simple observations?

What is the Bedford Level experiment and what does it prove?

How accurate are these curvature calculations?

📚 Official Data Sources

⚠️ Disclaimer

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