Distance Calculator

Three different distances behind one question

Ask how far it is from one place to another and there are at least three answers. The straight line through the air follows the curve of the planet and is the shortest path on the surface, which is the great circle. The road distance follows streets and is longer, often by a fifth or more. The travel time depends on the speed you can hold, not on the distance at all.

This page answers the first one from coordinates, and converts the answer into miles, kilometers and nautical miles. It also turns a distance into hours at whatever average speed you enter, which is the number people usually want when they ask about a trip.

The gap between the first two rows is the point. A straight line is the floor for the trip, never the road distance, and treating it as one is why a 2,445 mile measurement can sit next to a three day drive.

The haversine formula, step by step

Coordinates come as latitude and longitude in degrees. The formula needs radians, so multiply by 0.0174533 first. Then: a = sin squared(dLat / 2) + cos(lat1) x cos(lat2) x sin squared(dLon / 2), c = 2 x asin(square root of a), and d = R x c, where R is the radius of the sphere the Earth is being treated as.

The radius matters more than people expect. The Earth Fact Sheet from the National Aeronautics and Space Administration lists an equatorial radius of 6,378.137 km and a volumetric mean radius of 6,371.0 km, and using one instead of the other moves a 2,400 mile answer by a few miles. This calculator uses 6,371.0088 km, the mean radius, because a mean sphere keeps errors small in both directions and at all latitudes.

The cos(lat1) x cos(lat2) factor is what makes longitude shrink as you move toward the poles. One degree of longitude is 111.2 km at the equator and about 85.2 km at 40 degrees north, while one degree of latitude stays near 111.2 km everywhere. Ignore that factor and a trip across the United States comes out far too long.

Worked example: New York to Los Angeles

New York sits at 40.7128 north, 74.0060 west; Los Angeles sits at 34.0522 north, 118.2437 west. West longitudes are negative, so the longitudes entered are -74.0060 and -118.2437. Drop a minus sign and the calculator measures a path across the Atlantic and Asia instead, which returns about 11,600 km.

With the signs correct the answer is 3,935.8 km, which is 2,445.6 miles or 2,125.1 nautical miles. The initial bearing is 273.7 degrees, a hair north of due west, and that bearing changes along the route: a great circle from New York to Los Angeles starts nearly west and finishes nearly west-southwest because the route bends north in the middle.

Published city-pair figures differ from these by a few miles because they are measured between different points: an airport, a city hall, a geographic center. For any of the rows above, a ten mile difference between two sources is normal, and a hundred mile difference means one of them is measuring a road route.

Distance from speed and time

When the trip is already under way, the question flips: distance = speed x time. Sixty miles per hour for 2.5 hours covers 150 miles, which is 241.4 km. The same formula backward gives the time: 2,445.6 miles at 60 miles per hour is 40.76 hours, and that is driving time on a straight line with no stops.

Average speed is the whole input here, and it is always lower than the posted limit. Fuel stops, traffic, and the last few miles off the highway pull a 65 mph limit down to something closer to 55 mph over a long day, which turns 37.6 hours into 44 hours. The calculator returns the arithmetic; the average speed is yours to defend.

Miles, kilometers and nautical miles

The international mile is defined as exactly 1,609.344 meters, a definition fixed by international agreement in 1959 and listed in the National Institute of Standards and Technology guide to SI units. The nautical mile is exactly 1,852 meters, which is one minute of latitude, and that is why it divides evenly into degrees on a chart.

The neat row is the last one. A degree of latitude is 60 nautical miles by definition, so a route that runs 600 nautical miles north covers ten degrees of latitude. The circumference of that mean sphere is 2 x 3.14159 x 6,371.0088 = 40,030.2 km, and 40,030.2 / 360 = 111.195 km per degree, which is 69.09 miles. That single number covers most mental estimates: 40 degrees of latitude is about 4,448 km.

Coordinates, and the mistakes that break them

Latitude runs from 90 south to 90 north and longitude from 180 west to 180 east. Decimal degrees are the format this page wants, and the sign convention is the part that goes wrong: the western hemisphere is negative longitude and the southern hemisphere is negative latitude. New York is 40.7128, -74.0060, not 40.7128, 74.0060.

Degrees, minutes and seconds convert by dividing minutes by 60 and seconds by 3600. A coordinate written 40 degrees 42 minutes 46 seconds north becomes 40 + 42 / 60 + 46 / 3600 = 40.7128. Keeping the north and west labels in the text and dropping the signs is the most common error, and it produces a plausible number for a place nobody asked about.

A second source of error is mixing formats in one field: 40.7128 in one box and 34 3 8 in the next. Enter one format for both points. If a GPS device shows 40.712800 N and 74.006000 W, the two numbers to type are 40.7128 and -74.006.

When a straight line is the wrong answer

The great circle is the right number for a flight path, a satellite link budget, or a radio line of sight, and it is the wrong number for a road trip, a delivery route or a pipeline. Roads bend because of terrain, property lines and the grid they were laid on, and the ratio between road and straight line is usually between 1.15 and 1.4 in flat country, worse in mountains.

Water adds another step. A ferry route has to reach a terminal, so the nautical distance between two ports on opposite shores is a starting point, not the crossing. Weather routing can add a tenth to an ocean passage on its own.

Two pages on this site handle the neighbors of this question. The drive time calculator works from a road route and an average speed, which is what a trip plan needs. The parking distance calculator measures the geometry of a parking space and how far a car can be from a curb, which is a survey question about a few feet, not a route between towns. This page sits between them: two points, any distance, no roads and no parking lines.