How to find the midpoint between two coordinates

Why averaging the coordinates is wrong

Adding two latitudes and dividing by two feels natural, and for points a few miles apart it works. Over longer distances it fails because the shortest path between two points on a sphere is a great circle, and in the Northern Hemisphere a great circle bows toward the pole compared with a line of constant latitude. The true midpoint of an east-west trip therefore sits north of the averaged point.

Sacramento sits at 38.577° N, 121.494° W and Boston at 42.358° N, 71.064° W. Averaging gives 40.47° N, 96.28° W. The spherical midpoint is 43.32° N, 97.04° W, 2.85 degrees farther north. On a trip of 2,625 miles, the error is 201 miles.

The midpoint formula

Convert both points to radians. Call the latitudes φ1 and φ2, the first longitude λ1, and the difference in longitude Δλ. Then:

Bx = cos φ2 · cos Δλ
By = cos φ2 · sin Δλ
φm = atan2(sin φ1 + sin φ2, √((cos φ1 + Bx)² + By²))
λm = λ1 + atan2(By, cos φ1 + Bx)

The formula works by treating each point as a vector from the center of the Earth, adding the two vectors, and reading the direction of the sum as a new latitude and longitude. Because the sum points exactly between the two input vectors, the result lies on the great circle and is equally far from both ends.

Convert φm and λm back to degrees and normalize the longitude to the range from -180 to 180. That last step matters for trips that cross the 180th meridian, where a raw average of longitudes can land on the wrong side of the planet.

Averaged vs true midpoints for eight capital pairs

The gap grows with distance and with how far the trip runs east to west. Austin and Saint Paul, a mostly north-south pair, move by 12.7 miles, while Sacramento and Boston move by 201.

How to check a midpoint

A correct midpoint is the same straight-line distance from both ends, and those two distances add up to the full trip. Run the midpoint and each endpoint through the crow flies distance calculator. For Sacramento and Boston each half measures 1,312.4 and 1,312.4 miles. The averaged point fails this test: it is 1,345.6 miles from Sacramento and 1,308.3 from Boston, and the two legs add up to more than the direct trip.

Straight-line midpoint vs driving midpoint

The geometric midpoint is where two people would meet flying straight toward each other. Drivers meet at the halfway point of the road route, which can be far from the straight-line midpoint when highways detour around mountains, lakes or borders. Use the geometric answer to pick a region, then check drive times from both sides.

The halfway point tool below computes the spherical midpoint and lists towns near it, so you get a real place name instead of a point in a field.

Midpoints of three or more places

For a group, convert every point to a unit vector, add them all, and convert the sum back to latitude and longitude. This gives the geographic center of the group. Weight each vector by headcount if some places send more people. The geographic center finder does this for any list of points, and the center of population guide applies the same idea to every county in a state.

Why the midpoint looks off center on a map

Plot the true midpoint of Sacramento and Boston on a web map and it appears north of the straight line you would draw with a ruler. That is a property of the map, not the math. Web maps use the Mercator projection, which draws lines of constant compass heading as straight lines, while the shortest path bows toward the pole. The midpoint sits on the curved great circle, so on a flat map it looks shifted. The great circle route map draws the curved path with its midpoint so you can see both lines at once. For a pair a few miles apart, both lines overlap and the simple average is good enough.