Flat vs. Round Earth Calculator

What Is Earth Curvature and Why Does It Matter?

Earth curvature describes how the surface of a spherical Earth curves away from a straight horizontal line at any given distance. Because Earth is a sphere with a mean radius of approximately 6,371 km, the surface drops below a tangent line at a predictable rate that can be measured, calculated, and compared against observations. This curvature has practical consequences for navigation, telecommunications, aviation, photography, and the long-standing public debate about whether the Earth is flat or spherical.

The curvature effect is small at everyday distances but grows quickly. At 10 km, the surface drops about 7.8 m below a horizontal line. At 100 km, the drop is approximately 785 m. The European Space Agency's orbital photography archive documents the visual curvature clearly from altitudes above 10,000 m. This calculator uses the exact cosine formula rather than the common 8 inches per mile squared approximation, which becomes inaccurate beyond about 100 km.

The Curvature Formulas Explained

Three separate quantities describe Earth curvature at a given surface distance d:

Surface drop: h = R x (1 - cos(d/R)), where R = 6,371,000 m. This gives the vertical distance between the Earth's surface at distance d and the tangent plane at the observer. It equals the "bulge" seen from the side.

Horizon distance: d_h = sqrt(2 x R x e), where e is the observer eye height in metres. A person standing with eyes 1.7 m above sea level sees a horizon at approximately 4.65 km. Climbing to 100 m eye height extends the horizon to about 35.7 km. The formula uses the Pythagorean relationship between the observer, the horizon point, and the Earth's centre.

Hidden height: The portion of a distant target hidden below the horizon. Once the target lies beyond the horizon distance, the surface continues to curve and progressively hides the lower part of the target. The hidden height equals the surface drop from the horizon to the target's position. This is why ships disappear hull-first before their masts, and why the bottom floors of distant city skylines are hidden while upper floors remain visible.

All three quantities are modified by atmospheric refraction. Standard atmospheric conditions bend light rays slightly downward, effectively increasing the apparent Earth radius by a factor of 7/6. This calculator displays results both with and without this correction.

Globe vs. Flat Earth: What Each Predicts

The core distinction between the two models produces testable, measurable differences:

A spherical Earth predicts that at 50 km distance, a target at sea level will have its lower 49 m obscured by curvature (with standard refraction, roughly 43 m). This is precisely why the bottom of the Chicago skyline is hidden when viewed from the Michigan shore at roughly 80 km across Lake Michigan, while the upper floors of the Willis Tower remain visible. The Metabunk curvature calculator, built by science communicator Mick West, has been used to verify thousands of such observations with consistent accuracy.

A flat Earth predicts zero hidden height at any distance, because there is no curvature to hide anything. Under a flat Earth model, the entire Chicago skyline would be visible from Michigan at any observer height above sea level. It is not. Atmospheric refraction can extend visibility somewhat, but it cannot reduce a 49 m hidden height to zero. The side-by-side panel in this calculator makes this contrast explicit for any distance and observer height you enter.

The 8 inches per mile squared approximation (often quoted in flat Earth discussions) derives from the leading term of the cosine formula: h approximately equals d squared divided by 2R. In imperial units this gives 7.98 inches per mile squared. It is accurate to within 1% up to about 50 km but diverges at longer distances. This calculator always uses the exact formula.

Atmospheric Refraction and Why Objects Appear Further Than Geometry Predicts

Standard atmospheric refraction is the bending of light as it passes through air of varying density. Cold dense air near the surface has a higher refractive index than warmer air above, so light rays curve slightly downward rather than travelling in perfectly straight lines. The result is that an observer can see approximately 7% further than pure spherical geometry would allow.

The refraction effect is modelled by using an effective Earth radius of R_eff = R x 7/6 = 7,433 km, which gives a good approximation under standard atmospheric conditions (temperature lapse rate of approximately 6.5°C per kilometre). This is the correction used by this calculator when refraction is enabled.

Refraction varies with weather. Temperature inversions, where warm air sits above cold air near the surface, dramatically increase refraction and can create looming effects where normally-hidden objects become temporarily visible. This is the actual scientific explanation for instances where distant city skylines or ships reappear in unusual atmospheric conditions, not a flat Earth. The Journal of Geophysical Research Atmospheres published a 2021 study documenting extreme refraction events over Lake Michigan that caused the Chicago skyline to appear to float above the horizon, consistent with spherical-Earth models under anomalous refraction, not with a flat surface.

Famous Observations Explained by This Calculator

Several observations frequently cited in flat Earth discussions have straightforward spherical-Earth explanations:

• Chicago skyline from Michigan (80 km): At observer eye height 2 m, the globe predicts approximately 378 m hidden height. The Willis Tower is 442 m tall, so about 64 m of upper floors should be visible. This matches photographs taken from Romainian Park, Illinois, and was verified in a physics experiment context using this exact calculation.
• Bedford Level experiment (1838): Samuel Rowbotham claimed to see a boat travel 10 km along a canal without disappearing below the horizon, "proving" flatness. Later investigations showed the observer's eye was very close to the water surface where refraction is strongest, and that repeating the experiment with a raised observer eye height produced the expected curvature disappearance.
• Visibility of mountain peaks at long distance: Mountains with significant elevation remain visible at distances where a sea-level object would be fully hidden. Enter the mountain's height as the target height in this calculator to see precisely what fraction should remain visible.
• Airline flight paths: Flights between Southern Hemisphere cities (Sydney to Johannesburg, Buenos Aires to Cape Town) follow great circle routes that make sense only on a globe. On a flat Earth map, these routes would curve wildly north, adding thousands of kilometres. Flight time data is consistent with globe geometry to within normal air-traffic variation.

How to Use the Line-of-Sight Mode

The "Can A See B?" mode determines whether two positions at specified heights have an unobstructed line of sight across a spherical Earth. Enter the surface distance between them, the observer's eye height, and the target height. The calculator computes the hidden-height threshold and compares it to the target height:

• If the target height exceeds the hidden height, the top of the target is visible. The margin shows how many metres of the target are visible above the curvature threshold.
• If the target height is less than the hidden height, the target is fully obscured. The margin shows how much taller the target would need to be to become visible.

This mode is useful for radio tower placement (line-of-sight radio links require clearance above the curvature and Fresnel zone), maritime navigation (how far offshore can a lighthouse be seen?), and photography (will a distant mountain peak be visible from a planned viewpoint?). Atmospheric refraction can be toggled to compare worst-case (no refraction) and typical-condition (standard refraction) scenarios.

Common Errors When Calculating Earth Curvature

Several recurring mistakes produce incorrect curvature predictions:

• Using the 8 inches per mile squared approximation at long distances. This formula is an approximation valid only at short ranges. At 200 km it underestimates the drop by several percent. This calculator always uses the exact cosine formula.
• Ignoring observer eye height. The horizon distance and hidden-height calculations both depend critically on observer eye height. A person lying flat at sea level (eye height approximately 0.1 m) has a horizon under 1.1 km. Standing with eyes at 1.7 m extends this to 4.65 km. Flat Earth claimants who photograph distant objects from elevated positions without accounting for their height consistently underestimate curvature effects.
• Ignoring atmospheric refraction. Refraction accounts for roughly 7% of visible distance extension under standard conditions. Not including it leads to predictions that are systematically too pessimistic about visibility range.
• Confusing drop, bulge, and hidden height. Drop is measured vertically from the tangent plane to the surface. Hidden height is the portion of a target object below the line of sight from the observer. They are related but not the same quantity. This calculator shows all three clearly labelled.