Three observers, far apart, agree on one instant. Each faces true north and writes down two numbers: the compass bearing of the Sun, and its angle above the horizon. Six numbers in total. No travel, no equipment beyond a phone compass and a protractor.
Those six numbers are enough to decide the shape of the Earth. What settles it is arithmetic, and anyone with a calculator can check it.
Three people. One agreed instant, to the minute, in UTC. Each one does the following.
That is the entire experiment. Six numbers, plus three positions and one timestamp. The observers do not need to agree about anything except the moment.
On a globe the Sun is roughly 150 million kilometers away, which is far enough that its rays arrive at Earth effectively parallel. There is one point on the surface where the Sun is straight overhead, called the subsolar point, and every observer’s reading follows from where they stand relative to it.
Here dec is the Sun’s declination and H is the hour angle, the observer’s longitude difference from the subsolar meridian. The bearing follows from the same two angles. One Sun position, computed once, produces every observer’s pair of numbers. There is no per-observer adjustment available, because the model does not have one to give.
On the flat model the Sun is a local light circling above the disc at some height. Take an observer’s measured altitude and the horizontal distance from that observer to the point on the map beneath the Sun. Those two quantities fix the Sun’s height by simple trigonometry:
Run that for each observer in turn. Each one yields a height. There is one Sun, so the three answers must agree. The model has no free parameter left to rescue them, because the Sun’s ground position is already pinned by the bearings.
This is not a constructed case. Three observers agreed on 24 November 2025 at 2:47:13 in the afternoon, Eastern time, which is 19:47:13 UTC. At that instant the Sun stood over 20.7° S, 120.1° W, in the South Pacific.
| Observer | Bearing, computed | Altitude, computed | Shown by the flat-Earth app |
|---|---|---|---|
| Phoenix, Arizona | 189.2° | 35.3° | 190 SSW, alt 35° |
| Bristol, Virginia | 218.4° | 22.4° | 218 SW, alt 23° |
| Victor Harbor, South Australia | 113.3° | 3.3° | 105 ESE, alt 15° |
Two of the three agree with the sky to within a degree. Phoenix and Bristol come out right on both numbers, which is what a globe calculation does for any observer anywhere. Read only those two and there is no argument to be had.
The direction of the error is the part that matters. Phoenix sits about 6,100 km from the Sun’s position as drawn on the flat map. Victor Harbor sits about 20,300 km from it, more than three times as far. The flat model is nearly right near the middle of its map and badly wrong out at the rim, which is exactly where a projection has to stretch the southern hemisphere to fit it onto a disc.
Run the height calculation on the two northern readings and they imply a Sun about 4,300 km up from Phoenix and about 3,600 km up from Bristol. Those are close enough that the difference could be waved away as measurement slop. With two observers there is no experiment. Take their average as the model’s best answer, 3,900 km, and ask what altitude it predicts at Victor Harbor: 11°. The sky says 3.3°.
So the three readings cannot be reconciled. Two of them fix the Sun’s height between them and the third then disagrees with that height by a margin no instrument error covers. That is what an over-determined problem looks like when the model is wrong.
Perspective. Perspective changes the apparent size of an object and the angle it subtends. It does not change the direction light arrives from, and it cannot put an object below a horizon that has nothing to hide behind. A lamp above a plane is visible from everywhere on that plane.
Refraction. Air bends light by fractions of a degree near the horizon and by essentially nothing at high altitudes. The discrepancy at Victor Harbor is nearly twelve degrees, and it is worst where the air is thinnest along the line of sight rather than thickest. Invoking refraction to close a gap that size requires an atmosphere nobody has measured and that would distort every other observation ever made.
A different map. The figures above use the north-polar azimuthal equidistant projection, which is the map flat-Earth models actually use. Any other flat map changes the distances and therefore changes the implied heights, but it cannot make three inconsistent answers agree. The inconsistency comes from the geometry, not from the choice of projection.
The Sun is closer than claimed. That is not an objection, it is the concession. The experiment measures the height. Three observers return three different values for it.
The experiment costs nothing and can be run by three friends in an afternoon. A compass, a protractor, and an agreed time. The reason it does not get run often is not that it is hard.