Learn & Understand

Measuring the Planet With a Stick and a Shadow

Calculating the distance between two points on Earth assumes we know the size of the Earth itself — and remarkably, that was worked out over two thousand years ago by a Greek scholar using little more than a stick, a shadow, and a brilliant piece of reasoning. The story of Eratosthenes is where the measurement of our world began.

A Shadow in Two Cities

Eratosthenes, a scholar in ancient Alexandria, learned that at noon on the summer solstice in a city to the south, the Sun shone straight down a well, casting no shadow — it was directly overhead. Yet at the same moment in Alexandria, a vertical stick did cast a shadow. If the Earth were flat, the Sun's rays would strike both places at the same angle. The difference could mean only one thing: the surface curved.

From Angle to Circumference

By measuring the angle of the Alexandria shadow, Eratosthenes found it was a small fraction of a full circle. Knowing the distance between the two cities, he reasoned that this distance corresponded to that same fraction of the Earth's whole circumference — and simply scaled up. With one proportion, he estimated the size of the entire planet, astonishingly close to the modern value.

Why the Earth's Size Matters for Distance

Every calculation of distance from latitude and longitude relies on the Earth's radius — the very figure Eratosthenes first estimated. Because the planet is a sphere, distances are measured as arcs along its curved surface, and the radius sets the scale that turns an angular separation between two coordinates into a real distance in kilometers or miles.

The Curved-Surface Correction

This is also why distance on Earth cannot be found with flat-plane geometry. A degree of longitude spans a different ground distance near the poles than at the equator, so the calculation must account for the sphere. The standard method — spherical trigonometry applied to two coordinate pairs — is a direct descendant of the insight that the world is round and measurable.

Measuring the Distance

To compute the distance between two coordinates, use the Distance Between Coordinates Calculator. Find the direction to travel with the Compass Bearing Calculator, or the shortest flight path with the Great Circle Distance Calculator.