Pythagorean equirectangular distance

This capability estimates how far two points on Earth are from each other by treating the map as a local flat plane. It converts the difference in latitude and longitude into metres using a mean-latitude cosine scale, then applies the Pythagorean theorem to those east-west and north-south offsets. The result is an approximate ground distance in metres and kilometres, plus the projected deltas so you can audit the intermediate steps. The algorithm is fully deterministic: no network calls, no random sampling and no clock. Use it when you need a quick, cheap distance for nearby points, batch ranking, or teaching the equirectangular projection without the cost of a full great-circle formula.

How the Pythagorean equirectangular formula works

On a sphere, true shortest paths are great circles, but for many product and analysis tasks a flat local approximation is enough. This tool uses the classic equirectangular projection: convert both latitudes and longitudes to radians, take the difference in latitude as the north-south arc on a sphere of fixed mean radius, and scale the longitude difference by the cosine of the average latitude so that east-west metres shrink near the poles. Those two metre offsets become the legs of a right triangle.

The hypotenuse is the reported ground distance. We use a mean Earth radius of six million three hundred seventy-one thousand metres, the same constant many educational and GIS primers use for this method.

Valid coordinates and when a value is refused

Send decimal degrees on the WGS84 globe. Latitude fields lat1 and lat2 must sit between minus ninety and ninety inclusive. Longitude fields lon1 and lon2 must sit between minus one hundred eighty and one hundred eighty inclusive.

Out-of-range latitudes or longitudes fail the same way with a message that names the offending field. Prefer stable decimal numbers rather than degree-minute-second text or grid references; convert those formats before you call. The params example uses a short Manhattan pair so the probe and documentation stay realistic.

For identical points the distance is zero. Crossing the antimeridian with raw longitude differences can inflate the result; for paths that wrap the date line, normalise longitudes first or use a great-circle method instead.

When to use this approximation versus a spherical formula

Choose the Pythagorean equirectangular distance when points are relatively close, when you need maximum speed over maximum accuracy, or when you are teaching or debugging map projections and want intermediate metre deltas. Delivery radius checks inside a city, clustering of nearby sensors, rough travel-time heuristics and UI tooltips all fit well. For intercontinental routes, polar paths, or navigation where a few percent error matters, prefer a haversine or Vincenty-style great-circle distance instead: those methods account for Earth curvature more carefully.

Error grows with separation and with latitude extremes because the flat projection ignores meridional convergence beyond the mean-latitude scale factor. Always document the method when you store results so later audits know you used an approximation. Invalid inputs never consume balance, so you can validate coordinates aggressively before batch jobs.