Earth Curvature
Find how much of a distant object is hidden behind Earth's curve using d = √((R+h)²−R²). Free physics tool with unit conversion, charts, and step-by-step breakdowns for sailors, photographers, and surveyors.
About This Calculator
The Earth Curvature Calculator helps you determine how much of a distant object is obscured by the curvature of the Earth. Whether you are a sailor estimating lighthouse visibility, a photographer planning landscape shots, a hiker gauging mountain views, a surveyor performing fieldwork, or a student learning about Earth's spherical geometry — this tool provides instant, accurate results with unit conversion and visual charts.
The calculator uses basic spherical geometry and the Pythagorean theorem. Given your observer height (eye level above sea level) and the distance to a distant object, it computes: (1) your distance to the horizon using d = √((R + h)² − R²), and (2) the height of the object hidden below the horizon using x = √(a² − 2ad + d² + R²) − R. Here R = 6,371 km is Earth's mean radius, h is observer height, a is horizon distance from the observer, and d is distance from the observer to the object.
When the object is within your horizon distance, zero height is obscured — you can see it fully. Beyond the horizon, the obscured height increases non-linearly with distance. For example, at 1.75 m eye level, a ship 10 km away has about 2.5 m of its hull hidden; at 50 km, over 180 m is blocked. The interactive Curvature Chart visualizes this relationship across distances from 1 to 200 km.
Regional Notes: Earth's radius varies slightly — 6,378 km at the equator and 6,357 km at the poles. This calculator uses the mean radius of 6,371 km, accurate within ±0.3% worldwide. For metric users (India, Europe, and most countries), enter height in meters and distance in km. For the US, switch to feet and miles with the unit toggles. The UK uses both metric and imperial — the calculator supports both seamlessly.
Important: This calculator shows the geometric (no-refraction) obscured height. Atmospheric refraction bends light around the planet, letting you see about 8% farther than the geometric horizon under typical conditions. The actual visible horizon distance and obscured height may differ slightly from computed values depending on temperature gradients, humidity, and air pressure.
Frequently Asked Questions
What is the Earth's curvature?
Earth's curvature refers to the way the planet's spherical surface curves away from a straight line. Due to Earth's radius of about 6,371 km, the surface drops approximately 8 inches per mile. This curvature causes distant objects to disappear below the horizon as they move farther away.
How far can I see before the Earth curves?
At sea level with eyes at 1.75 m height, your horizon is about 4.7 km (2.9 miles) away. From a height of 100 m, you can see approximately 35.7 km. The formula is d = √((R + h)² − R²) where R is Earth's radius (6,371 km) and h is your height above sea level.
How much of a distant object is hidden by the curvature?
The hidden height depends on your observer height and the distance to the object. For example, at 1.75 m eye height, a 10 km distant object has about 2.5 m obscured. At 50 km distance, over 180 m is hidden. The obscured height equals √(a² − 2ad + d² + R²) − R where a is horizon distance, d is distance to object, and R is Earth's radius.
Does atmospheric refraction affect what I see?
Yes, atmospheric refraction bends light slightly around the Earth's surface, allowing you to see slightly farther than the geometric horizon. On average, refraction increases horizon distance by about 8%. This calculator provides geometric (no refraction) values, which are slightly conservative.
Can I use this for non-Earth celestial bodies?
This calculator uses Earth's mean radius of 6,371 km. For other celestial bodies like the Moon (radius 1,737 km), Mars (3,390 km), or Jupiter (69,911 km), the horizon distance and obscured height will differ significantly. You can approximate by replacing Earth's radius with the target body's radius, but atmospheric effects will also differ. For dedicated multi-body calculations, see our Flat vs. Round Earth Calculator for comparative experiments.
Is 8 inches per mile accurate for Earth's curvature?
Approximately, yes. The drop due to curvature is roughly 8 inches per mile squared. For a more precise calculation, the exact formula is h = R − √(R² − d²), where R = 6,371 km and d is the distance. At 1 mile (1.609 km), the drop is about 8 inches (20 cm). At 2 miles, it's about 32 inches.
How do you calculate the distance to the horizon?
The distance to the horizon is calculated using the Pythagorean theorem: d = √((R + h)² − R²), where R is Earth's radius (6,371,000 m) and h is the observer's height above sea level in meters. For a 1.75 m tall person at sea level, the horizon is approximately 4.7 km away.
Why can't I see the bottom of a distant ship?
As a ship moves farther away, the Earth's curvature blocks progressively more of its hull. At 10 km distance from a 2 m eye height, roughly 4 m of the ship's bottom is hidden. This phenomenon is one of the oldest observational proofs of Earth's spherical shape.