Distance Calculator

The distance formula in 2D, and the Pythagorean theorem

The equation for distance between two points on a flat plane, also called the distance formula, is an application of the Pythagorean theorem. Take two points, (x1, y1) and (x2, y2). The horizontal gap between them, x2 minus x1, and the vertical gap between them, y2 minus y1, form the two legs of a right triangle. The straight-line distance between the points is the hypotenuse of that triangle, so the same a² + b² = c² relationship applies, rearranged into the distance formula. That rearrangement is how to find distance between any two coordinates in the 2D Distance Calculator above: enter both points, press Calculate, and the calculator applies this same equation.

Worked example. Take the points (0, 0) and (3, 4). The horizontal leg is 3 minus 0, which is 3. The vertical leg is 4 minus 0, which is 4. Squaring and adding those legs gives 3² + 4² = 9 + 16 = 25, and the square root of 25 is exactly 5. That 3-4-5 relationship is the classic right triangle used to introduce the Pythagorean theorem in the first place, and it is exactly what the 2D Distance Calculator above computes when you enter those same four numbers and press Calculate.

Extending the formula into three dimensions

The 3D version of the formula, sometimes searched as the distance formula in 3 space, adds one more axis, z, for depth, alongside x and y. Each point becomes (x1, y1, z1) and (x2, y2, z2), and the formula gains one more squared difference under the square root:

Geometrically, this is the same right-triangle idea applied twice: once to reach the point directly below or above the target in the xy-plane, and once more using that intermediate distance and the z difference as the two legs of a second right triangle.

Worked example. Take the points (0, 0, 0) and (2, 3, 6). The three differences are 2, 3, and 6. Squaring and adding them gives 2² + 3² + 6² = 4 + 9 + 36 = 49, and the square root of 49 is exactly 7. Enter those same six numbers into the 3D Distance Calculator above, press Calculate, and the calculator returns the same clean answer.

Great-circle distance between two coordinates: the haversine formula

Use the Distance Based on Latitude and Longitude calculator to type two latitude and longitude points, or the Distance on Map calculator to click them directly on an interactive map. Enter latitude and longitude for point 1 and point 2 and press Calculate, or click once on the map for point A and again for point B. The Distance Based on Latitude and Longitude calculator shows its answer after Calculate is pressed, and the Distance on Map calculator updates immediately after each click, both in kilometers or miles using the haversine formula for great-circle distance across the Earth's surface, the distance travelled “as the crow flies”. Unlike the 2D Distance Calculator and 3D Distance Calculator above, latitude and longitude describe a real position on Earth, so this result is a genuine geographic distance rather than an abstract coordinate-plane one. In other words, these two calculators work as an as the crow flies distance calculator, or simply a crow flies distance calculator, whenever the straight-line figure between two real places, not a driving route, is what is needed.

Worked example. New York City sits at roughly 40.7128°N, 74.0060°W, and London at roughly 51.5074°N, 0.1278°W. Entering those four numbers into the Distance Based on Latitude and Longitude calculator and pressing Calculate returns a great-circle distance of about 5,570 kilometers, or 3,461 miles. That is the shortest possible path between the two cities if travel could go in a straight line through the air, not the distance a flight actually covers once real routes, wind and airspace are taken into account. Checked the other way, London to New York returns the same 5,570 kilometers, since great-circle distance does not depend on direction. A plain distance from NYC search almost always means exactly this: the distance to some other named city, which is what the Distance Based on Latitude and Longitude and Distance on Map calculators above work out for any two places you pick, not only London.

Latitude must fall between -90 and 90 degrees and longitude between -180 and 180 degrees, and the calculator rejects a value outside that range rather than returning a distance computed from a coordinate that cannot exist. Southern latitudes and western longitudes are entered as negative numbers, exactly as negative coordinates work in the 2D Distance Calculator and 3D Distance Calculator above. The most common mistake in this calculator is swapping latitude and longitude, since both are just plain numbers and a swapped pair does not always trigger that range check, for example 40 is a valid value for either one, so the result can look plausible while describing the wrong pair of points entirely.

Finding distance with the interactive map in this distance calculator

The Distance on Map calculator uses the same haversine formula, applied to two points you click instead of type, working as a distance between two points map rather than a form to fill in. The first click places point A, the second places point B, and every click after that moves point B, so a point can be adjusted without starting over. Both points’ coordinates appear underneath the map next to the distance between them, which is a fast way to check how far apart two visible places are, such as two cities, without knowing either one’s exact latitude and longitude in advance. It is still the same great-circle distance described above, not a driving route, so it will read lower than an actual road trip between the same two points.

Nothing about a click leaves your browser. The map tiles come from OpenStreetMap, but the two points you place and the distance calculated between them are never sent anywhere, the same privacy guarantee as the other three calculators.

Great-circle distance is not driving distance

This is worth stating plainly now that two of the four calculators compute real geographic distance. The Distance Based on Latitude and Longitude and Distance on Map calculators return the great-circle distance between two points on Earth, close to the measurement airlines and shipping routes are planned around at a large scale, not the distance a person would actually drive, walk or fly door to door. A road has to curve around mountains, coastlines, other buildings and the road network itself, so it is essentially always longer than the straight-line figure this calculator returns. Boston to New York, for example, is about 306 kilometers (190 miles) as the crow flies, shorter than the distance shown by a mapping app that is routing an actual road trip. San Francisco to Los Angeles is about 559 kilometers (347 miles) the same way, again shorter than the highway distance between the two cities.

The same straight-line math applies to plenty of other well-known pairs. Austin to Houston works out to about 236 kilometers (146 miles) as the crow flies. How far is Houston from Dallas the same way? About 362 kilometers (225 miles). Houston to New Orleans comes out to roughly 512 kilometers (318 miles). Further north, Boston to DC is about 634 kilometers (394 miles), and that same Boston to NYC hop, using the common shorthand for New York City, is the same 306 kilometers (190 miles) already worked out above. Every one of those figures is the great-circle line between the two cities’ coordinates, not a driving distance, so an actual road trip between any of these pairs will read higher on a mapping app.

The 2D Distance Calculator and 3D Distance Calculator are a separate case again: they measure an abstract coordinate plane or coordinate space with no connection to any real-world location at all, which is why typing latitude and longitude into those two calculators, instead of the Distance Based on Latitude and Longitude calculator, will not produce a meaningful result. For the full derivation of the haversine formula, multiple worked examples and how it compares with more precise ellipsoidal methods, see the detailed guide on great-circle distance.