Geographic Midpoint Calculator
Find the true geographic center between two GPS coordinates, with optional weighting for each location. Useful for choosing a meeting point, logistics hub, or central headquarters between multiple sites.
About this calculator
A simple average of latitudes and longitudes does not give the correct midpoint on a sphere — it fails near the poles and across the antimeridian. The correct method converts each coordinate to 3-D Cartesian (X, Y, Z) vectors on the unit sphere, computes a weighted average of those vectors, then converts back to latitude and longitude. For each point: X = cos(lat) × cos(lon), Y = cos(lat) × sin(lon), Z = sin(lat). Weighted averages are X̄ = (X₁·w₁ + X₂·w₂)/(w₁+w₂), and similarly for Ȳ and Z̄. The midpoint latitude is midLat = atan2(Z̄, √(X̄²+Ȳ²)) and midpoint longitude is midLon = atan2(Ȳ, X̄), both converted back to degrees. Equal weights (w₁ = w₂ = 1) give the pure geographic center; unequal weights bias the result toward the more important location.
How to use
Find the midpoint between London (51.51°N, −0.13°E) and New York (40.71°N, −74.01°E) with equal weights (1, 1). Step 1 — London vectors: X₁ = cos(51.51°)×cos(−0.13°) ≈ 0.6225, Y₁ ≈ −0.00141, Z₁ = sin(51.51°) ≈ 0.7822. Step 2 — New York vectors: X₂ = cos(40.71°)×cos(−74.01°) ≈ 0.2080, Y₂ ≈ −0.7275, Z₂ = sin(40.71°) ≈ 0.6521. Step 3 — averages: X̄ ≈ 0.4153, Ȳ ≈ −0.3645, Z̄ ≈ 0.7172. Step 4 — midLat = atan2(0.7172, √(0.4153²+0.3645²)) ≈ 50.9°N. The midpoint lies roughly over the North Atlantic near 51°N, 41°W — consistent with a transatlantic great-circle path.
Frequently asked questions
Why is averaging latitudes and longitudes directly not accurate for finding a geographic midpoint?
Latitude and longitude are angular coordinates on a sphere, not linear coordinates on a flat plane. Simply averaging them treats the Earth as flat, which introduces errors that grow with distance and become severe near the poles or when the two points straddle the antimeridian (±180° longitude). For example, the midpoint between two points at longitudes 170°E and 170°W is 0° by naive averaging, but the true midpoint is 180°. Converting to Cartesian vectors, averaging in 3-D space, then converting back correctly accounts for the Earth's curvature and always produces the shortest-path midpoint.
What does the location weight parameter do in the geographic midpoint calculator?
The weight parameter lets you bias the midpoint toward one location over another — useful when the two sites are not equally important. For instance, if you are locating a warehouse that will serve a city of 1 million (weight 10) and a town of 100,000 (weight 1), the optimal hub should be much closer to the city. Mathematically, the weighted Cartesian vectors are scaled by their respective weights before averaging, pulling the resulting midpoint proportionally toward the heavier location. Setting both weights equal to 1 gives the pure geographic center.
How can I use the geographic midpoint to plan a fair meeting location between two cities?
Enter the latitude and longitude of each city (easily found on Google Maps by right-clicking), set both weights to 1, and the calculator returns the midpoint latitude and longitude. You can then paste those coordinates into a mapping app to see the nearest town or landmark. For truly fair travel, also check that travel times are roughly equal — a midpoint over a mountain range or ocean may be geographically central but impractical. Adjust the weights if one party is traveling by a slower mode of transport to balance actual journey time.