Flat vs Round Earth Calculator

Calculate Earth's curvature, horizon distance, and visibility using scientific formulas.

Compare flat Earth vs round Earth models by calculating horizon distance, curvature drop, and visibility based on observer height and distance.

Examples

Click on any example to load it into the calculator.

Beach Observation

beach

A person standing on a beach looking for ships on the horizon.

Observer Height: 1.7 m

Target Height: 20 m

Distance: 15 km

Refraction Index: 1.08

Mountain Peak View

mountain

Viewing from a mountain peak to another mountain range.

Observer Height: 2000 m

Target Height: 3000 m

Distance: 50 km

Refraction Index: 1.0

Skyscraper Observation

building

Looking from one tall building to another across a city.

Observer Height: 100 m

Target Height: 150 m

Distance: 5 km

Refraction Index: 1.05

Airplane Horizon

airplane

Viewing the horizon from a commercial airplane at cruising altitude.

Observer Height: 10000 m

Target Height: 0 m

Distance: 350 km

Refraction Index: 1.0

Other Titles
Understanding the Flat vs Round Earth Calculator: A Comprehensive Guide
Explore the scientific evidence for Earth's curvature through practical calculations. This guide explains the physics behind horizon distance, curvature drop, and why these calculations prove Earth is round.

What is the Flat vs Round Earth Calculator?

  • Core Concepts
  • Scientific Basis
  • Why These Calculations Matter
The Flat vs Round Earth Calculator is a scientific tool that demonstrates Earth's curvature through mathematical calculations. It uses established physics formulas to calculate horizon distance, curvature drop, and visibility between two points on Earth's surface. These calculations provide irrefutable evidence that Earth is a sphere, not flat, by showing how the planet's curvature affects what we can see at different distances.
The Scientific Foundation
This calculator is based on well-established scientific principles. Earth has a radius of approximately 6,371 kilometers, and this curvature affects how far we can see. The horizon distance formula (√(2 × R × h)) is derived from basic geometry and has been verified through countless observations and experiments. When you stand on a beach and look out to sea, the reason you can't see ships beyond a certain distance is Earth's curvature, not because they 'fall off the edge.'
Why These Calculations Are Important
Understanding Earth's curvature is crucial for navigation, astronomy, and debunking flat Earth misconceptions. These calculations show that the 'flat Earth' model cannot explain basic observations like why ships disappear hull-first over the horizon, why the horizon appears curved from high altitudes, or why different stars are visible from different latitudes. The mathematical precision of these calculations provides concrete evidence that Earth is spherical.
The Horizon Distance Formula
The key formula used in this calculator is: Horizon Distance = √(2 × Earth's Radius × Observer Height). This formula tells us how far an observer can see to the horizon based on their height above sea level. For example, a person 1.7 meters tall can see approximately 4.7 kilometers to the horizon, while someone at 100 meters elevation can see about 36 kilometers. This relationship between height and visibility is impossible to explain with a flat Earth model.

Key Scientific Principles:

  • Earth's Radius: 6,371 km (established through centuries of measurements)
  • Horizon Formula: d = √(2Rh) where R is Earth's radius and h is observer height
  • Curvature Drop: The amount Earth's surface curves away over a given distance
  • Atmospheric Refraction: How air density affects light bending and visibility

Step-by-Step Guide to Using the Calculator

  • Input Parameters
  • Understanding Results
  • Interpreting the Data
Using this calculator effectively requires understanding what each input means and how the results prove Earth's curvature. Follow these steps to get the most accurate and meaningful results.
1. Observer Height - Your Viewing Position
Enter your height above sea level in meters. This could be your eye level if standing on the ground (typically 1.5-1.8 meters), or much higher if you're on a building, mountain, or airplane. The higher you are, the farther you can see to the horizon. This is because you're looking over more of Earth's curvature.
2. Target Height - What You're Looking At
This is the height of the object you're trying to see, also in meters. It could be a building, mountain, ship, or any other object. If you're looking at something at ground level, use 0. The calculator will determine if this target should be visible given Earth's curvature.
3. Distance - How Far Away
Enter the distance between you and the target in kilometers. This is the key variable that determines how much of Earth's curvature lies between you and your target. As distance increases, more of Earth's surface curves away from your line of sight.
4. Refraction Index - Atmospheric Effects
This accounts for how Earth's atmosphere bends light. A value of 1.0 means no refraction (perfect vacuum), 1.08 is typical atmospheric conditions, and 1.15 represents strong refraction. Atmospheric refraction can sometimes make objects visible that would otherwise be hidden by curvature.
5. Interpreting Your Results
The calculator provides several key results: Horizon Distance (how far you can see), Curvature Drop (how much Earth curves away over your specified distance), and Target Visibility (whether your target should be visible). If the curvature drop is greater than the target height, the target should be hidden by Earth's curvature.

Common Observer Heights:

  • Eye Level (Standing): 1.7 meters
  • Beach Level: 2 meters
  • Tall Building: 100-500 meters
  • Mountain Peak: 1000-4000 meters
  • Commercial Airplane: 10,000-12,000 meters

Real-World Applications and Evidence

  • Navigation and Surveying
  • Astronomical Observations
  • Everyday Proofs
These calculations aren't just theoretical - they have practical applications in navigation, surveying, and astronomy. They also provide everyday evidence that Earth is round.
Maritime Navigation
Sailors have used horizon distance calculations for centuries. The fact that ships disappear hull-first over the horizon, with their masts being the last thing visible, is impossible to explain with a flat Earth model. This calculator shows exactly why this happens - as distance increases, more of Earth's surface curves away, hiding the lower parts of objects first.
Surveying and Engineering
Surveyors must account for Earth's curvature when measuring large distances. Over long distances, Earth's curvature becomes significant enough to affect measurements. This calculator helps engineers and surveyors understand how much correction is needed for their instruments and calculations.
Astronomical Observations
The visibility of different stars from different latitudes on Earth is only possible if Earth is spherical. This calculator's principles help explain why Polaris (the North Star) is visible in the Northern Hemisphere but not in the Southern Hemisphere, and why different constellations are visible from different parts of the world.
Everyday Evidence
You can test these calculations yourself. Go to a beach or large body of water on a clear day. As ships approach, you'll see their masts first, then gradually more of the ship becomes visible. This is exactly what the calculator predicts - objects emerging from behind Earth's curvature as they get closer.

Historical Evidence:

  • Eratosthenes calculated Earth's circumference in 240 BCE using shadows
  • Ferdinand Magellan's crew circumnavigated Earth in 1522
  • Modern GPS satellites require spherical Earth calculations to function
  • Photographs from space clearly show Earth's spherical shape

Common Misconceptions and Flat Earth Claims

  • Debunking Myths
  • Scientific Rebuttals
  • Why Flat Earth Doesn't Work
Flat Earth proponents often make claims that seem plausible at first but fall apart under scientific scrutiny. Let's examine the most common misconceptions and why they're incorrect.
Myth: 'The Horizon Always Appears Flat'
This is a common flat Earth claim, but it's easily debunked. The horizon appears flat because you're looking at a very small portion of Earth's surface. From high altitudes (airplanes, mountains), the horizon's curvature becomes clearly visible. This calculator shows that even at ground level, the horizon is actually curved - it just appears flat due to the limited field of view.
Myth: 'Atmospheric Refraction Explains Everything'
While atmospheric refraction does affect visibility, it cannot explain all the observations that prove Earth's curvature. Refraction typically makes objects appear slightly higher than they actually are, but it cannot make objects visible that are completely hidden by curvature over large distances. The calculator includes refraction effects, but they don't change the fundamental spherical nature of Earth.
Myth: 'Water Always Finds Its Level'
Flat Earth proponents claim that water always finds its level, proving Earth is flat. However, this misunderstands how gravity works. Water does find its level - it follows the gravitational field, which points toward Earth's center. On a spherical Earth, this means water naturally curves to follow the planet's surface. The oceans are curved, not flat.
Why Flat Earth Models Fail
Flat Earth models cannot explain: why different stars are visible from different latitudes, why the sun's path changes with seasons, why ships disappear hull-first over the horizon, why the horizon appears curved from high altitudes, or why GPS and satellite communications work. The mathematical precision of this calculator's formulas provides concrete evidence that Earth is spherical.

Scientific Rebuttals:

  • Horizon curvature is visible from high altitudes and in photographs
  • Different star patterns visible from different latitudes prove spherical Earth
  • GPS satellites require spherical Earth calculations to function accurately
  • The sun's seasonal path changes are only possible on a spherical Earth

Mathematical Derivation and Examples

  • The Horizon Formula
  • Curvature Calculations
  • Practical Examples
The mathematical foundation of this calculator is based on simple geometry and trigonometry. Understanding these formulas helps explain why Earth must be spherical.
Deriving the Horizon Distance Formula
The horizon distance formula comes from the Pythagorean theorem applied to a right triangle formed by Earth's radius, the observer's height, and the distance to the horizon. If R is Earth's radius and h is observer height, then: (R + h)² = R² + d², where d is the horizon distance. Solving for d gives us: d = √(2Rh + h²). Since h is very small compared to R, we can approximate this as d = √(2Rh).
Calculating Curvature Drop
The curvature drop formula calculates how much Earth's surface curves away over a given distance. For a distance d, the curvature drop is approximately: drop = d²/(2R). This formula shows that curvature drop increases with the square of distance, meaning it becomes more significant over longer distances.
Atmospheric Refraction Effects
Atmospheric refraction bends light rays, making objects appear higher than they actually are. This effect is included in the calculator by using an effective Earth radius that's larger than the actual radius. The refraction index typically ranges from 1.0 (no refraction) to 1.15 (strong refraction).
Practical Calculation Examples
Let's work through an example: A person 1.7 meters tall standing on a beach. Using the formula d = √(2 × 6,371,000 × 1.7), we get d ≈ 4,650 meters or about 4.7 kilometers. This means they can see about 4.7 km to the horizon. If they're looking at a ship 10 km away that's 20 meters tall, the curvature drop at 10 km is approximately 7.8 meters, meaning the ship's hull should be hidden by Earth's curvature.

Mathematical Constants:

  • Earth's Radius (R): 6,371,000 meters
  • Gravitational Acceleration (g): 9.81 m/s²
  • Speed of Light (c): 299,792,458 m/s
  • Typical Refraction Index: 1.08 (varies with conditions)