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Flat vs. Round Earth Calculator

Rather than accept centuries of scientific consensus, some people claim Earth is flat—despite overwhelming observational evidence to the contrary. This calculator lets you run three repeatable experiments using simple equipment: timing a sunset from different heights, measuring stick shadows at distant locations, and observing how objects vanish bottom-first over water. Each method independently confirms what Eratosthenes proved around 240 BC: our planet is a sphere.

Last updated:

Creators Dominik Czernia, PhD
8,950 people find this calculator helpful

The Flat Earth Model and Modern Skepticism

Remarkably, surveys indicate a small but vocal population questions Earth's sphericity, often citing distrust of institutions rather than engaging with observable phenomena. The flat Earth model proposes a disk with the North Pole at its center, surrounded by an ice wall at the southern edge to prevent water from falling off into space.

This model contradicts thousands of years of recorded observation. Ancient Greek philosophers including Pythagoras and Aristotle recognized Earth's curvature from phenomena like:

  • Ships disappearing hull-first over the horizon
  • Varying star positions at different latitudes
  • Earth's round shadow on the Moon during lunar eclipses
  • Consistent shadow angles from the Sun at different locations

Modern flat Earth claims ignore these direct, replicable observations in favour of abstract conspiracy theories.

The Stick Shadow Method: Calculating Earth's Circumference

The ancient Greek mathematician Eratosthenes used a simple stick and geometry to estimate Earth's circumference to within a few percent of the modern value. By measuring shadow lengths at two locations separated by a known north-south distance, you can replicate his calculation.

The method assumes the Sun is so distant that its rays arrive nearly parallel at both locations. The angle each shadow makes with the stick differs slightly due to Earth's curvature. The greater the distance between your two measurement points, the more accurate your result.

θ₁ = arctan(shadow length₁ ÷ stick height)

θ₂ = arctan(shadow length₂ ÷ stick height)

Earth circumference = distance (θ₂ − θ₁) ÷ (2π)

  • θ₁, θ₂ — Shadow angles (in radians) measured at locations A and B
  • shadow length₁, shadow length₂ — Length of shadow cast by the stick at each location
  • stick height — Vertical height of the stick
  • distance — North-south distance between the two measurement locations

Common Pitfalls When Conducting These Experiments

Successful results require careful setup and timing. Avoid these frequent mistakes:

  1. Comparing shadows at different times of day — Shadow angles change continuously as the Sun moves across the sky. You and your partner must measure shadows at the exact same local solar time, or your angles will be meaningless. Use a clock app showing solar noon for your latitude.
  2. Measuring close locations for stick shadows — If you measure shadows only 50 miles apart, small measurement errors dominate your result. Aim for at least 500 miles of north-south separation. The further apart you are, the larger the angle difference and the more accurate your Earth circumference estimate.
  3. Wind and mirror effects near water — For the horizon-disappearing experiment, strong winds create ripples that distort your view. Morning is better than afternoon because heat shimmer (mirage effect) is minimal. Calm, clear conditions are essential for seeing the object vanish bottom-first.
  4. Timing the sunset-twice effect — You must move upward continuously as the Sun sets. A single jerky jump won't work—the Sun sets faster than you might expect. Use an escalator, hill, or building where you can walk steadily upward while watching the horizon.

Three Observable Proof Experiments

Experiment 1: The Double Sunset — Lie down and watch the sunset until the Sun fully disappears below the horizon. Stand up immediately. If Earth is round, you'll see a thin slice of the Sun reappear because your eyes are now higher and the horizon has moved away from you. On a flat plane, no second sunset would occur regardless of height.

Experiment 2: The Vanishing Ship — Stand on a beach with binoculars and watch a ship sail away. On a round Earth, the hull disappears first, followed gradually by the superstructure and mast. On a flat surface, the entire ship would shrink uniformly and disappear in the distance. This effect is undeniable and requires no equipment beyond your eyes.

Experiment 3: The Hidden Object — From a low vantage point near a large lake, observe a distant building or vehicle several kilometres away on the far shore. Lie down as far as possible while remaining at the water's edge. The bottom of the object vanishes behind Earth's curve. Stand up and the bottom reappears. The curvature literally hides the lower portion of distant objects.

Historical Context and Modern Understanding

The spherical Earth was established as scientific consensus by the 5th century BC, when Pythagoras and later Aristotle presented geometric arguments. By the time of Eratosthenes (240 BC), calculating Earth's radius was routine mathematics for scholars.

Medieval European scholars never seriously doubted the spherical model despite popular myth. Columbus's voyage proved sailing routes possible on a curved globe. The age of exploration, gravity, satellite imagery, and physics all reinforce the same conclusion: Earth is an oblate spheroid (slightly flattened at the poles).

Today, millions of people routinely observe Earth's curvature aboard aircraft at cruising altitude. Satellites transmit signals that depend on orbital mechanics around a sphere. GPS requires spherical geometry. The convergence of evidence across multiple independent disciplines makes the round Earth one of the most thoroughly confirmed facts in science.

Frequently Asked Questions

What does the stick shadow experiment actually measure?

This experiment measures the angle at which the Sun's rays strike a vertical stick at two different latitudes. Because Earth is curved, the same Sun casts shadows at different angles thousands of kilometres apart. By comparing these angles and knowing the distance between measurement locations, you calculate how much of Earth's circumference you've sampled. Eratosthenes's original estimate (around 40,000 km) was remarkably close to the modern polar circumference of 40,008 km, proving both the method and the spherical geometry work.

Why does an object disappear bottom-first over the horizon?

A curved surface blocks the line of sight to lower objects before blocking higher ones. Imagine standing on the shore looking at a boat far away: the curvature of the water's surface literally rises up between you and the boat's hull, hiding it from view. But you can still see the boat's cabin and mast above that curve. If Earth were flat, the boat would shrink uniformly in all directions as it receded, never disappearing bottom-first. This observation requires no instruments—just patience and clear weather.

How high do you need to be to see the double sunset effect?

Any height gain works in principle, but the effect becomes noticeable above about 10 metres (33 feet). At sea level on a flat horizon, standing from lying down gives roughly 1.7 metres of height gain, allowing you to see a sliver of sunset repeat. From a 100-metre building or cliff, the effect is much more dramatic—the Sun can stay visible for several extra minutes. Aircraft at 10,000 metres altitude can demonstrate this effect dramatically, though clouds and window reflections often interfere with observation from planes.

Would the stick shadow experiment work on a flat Earth?

No. If Earth were a flat disk, stick shadows would follow flat-plane trigonometry: the shadow length would depend only on the Sun's angle in the sky, which doesn't change with latitude on a flat model. You'd measure the same shadow angle everywhere if the Sun were far away, contradicting real observations. Alternatively, if you invoked a moving Sun low above the disk, the mathematics leads to absurd results (like the Sun being impossibly close or the disk impossibly large). The stick shadow method only yields sensible, consistent results when applied to a sphere.

Can you see Earth's curvature from a commercial airplane?

Yes, but with caveats. From a cruising altitude of 10,500 metres (35,000 feet), Earth's curvature becomes visible if conditions are perfect: a completely clear, cloud-free horizon and a wide field of view. Most airplane windows are small and show clouds, making this difficult in practice. From 15,000+ metres, curvature becomes unmistakable. However, even standing on Mount Everest (8,849 metres), curvature is not obvious—you need greater altitude and exceptional visibility. High-altitude balloons and rockets make the curvature obvious to anyone watching.

Why don't more people perform these experiments themselves?

Inertia and convenience. Most people accept evidence from reliable sources rather than repeat experiments personally. The stick shadow method requires travel, coordination with a distant partner, and careful measurement. The sunset and horizon experiments demand specific weather and geography. Yet the experiments are not difficult—any motivated person can replicate them. The real barrier is that these observations have been replicated millions of times over centuries, so performing them yourself adds no new knowledge. Skeptics who actually conduct these experiments consistently obtain results matching spherical geometry, never flat-Earth predictions.

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