Geographic Midpoint Calculator
Add two or more locations by address, coordinates, map click or your location, optionally with weights. Get the geographic midpoint (centre of gravity), the centre of minimum distance and the simple average, with the nearest address and the distance to every place.
Or coordinates in any format: 56.9496, 24.1052 · 56°56′58.6″N · N 56° 56.976′
40.712800, -74.006000 — New York
Or coordinates in any format: 56.9496, 24.1052 · 56°56′58.6″N · N 56° 56.976′
41.878100, -87.629800 — Chicago
Or coordinates in any format: 56.9496, 24.1052 · 56°56′58.6″N · N 56° 56.976′
25.761700, -80.191800 — Miami
Midpoint · Geographic midpoint (centre of gravity)
36.252618, -80.544100
36°15′09.4″N 80°32′38.8″W
Distance from the midpoint
| 1. New York | 468.8 mi |
| 2. Chicago | 543.3 mi |
| 3. Miami | 725.2 mi |
| Total distance | 1,737.2 mi |
All methods compared
| Geographic midpoint (centre of gravity) | 36.252618, -80.544100 | Σ 1,737.2 mi |
| Centre of minimum distance | 38.510070, -79.760042 | Σ 1,698.9 mi |
| Simple average of latitude/longitude | 36.117533, -80.609200 | Σ 1,741.6 mi |
How it is calculated (with your numbers)
- Each location → 3D unit vector: x = cos φ · cos λ, y = cos φ · sin λ, z = sin φ
- 1: (40.7128°, -74.0060°) → x = 0.208854, y = -0.728647, z = 0.652268
- 2: (41.8781°, -87.6298°) → x = 0.030792, y = -0.743930, z = 0.667548
- 3: (25.7617°, -80.1918°) → x = 0.153419, y = -0.887446, z = 0.434629
- Weighted average of the vectors: x̄ = Σ wᵢxᵢ / Σ wᵢ = 0.131022, ȳ = -0.786674, z̄ = 0.584815 (Σ w = 3)
- Back to latitude and longitude: λ = atan2(ȳ, x̄) = -80.544100°, φ = atan2(z̄, √(x̄² + ȳ²)) = 36.252618°
- Centre of minimum distance: start at the midpoint and test points 8 directions around it, halving the step until it is under 1 cm; keep any point with a smaller Σ w·d → 38.510070, -79.760042
- Simple average of latitude/longitude: φ̄ = Σ wᵢφᵢ / Σ wᵢ, λ̄ = Σ wᵢλᵢ / Σ wᵢ → 36.117533, -80.609200
Map
Search and addresses: Nominatim. Map data © OpenStreetMap contributors (ODbL). Queries are sent from your browser only when you press a button or click the map. © OpenStreetMap
About this tool
Geographic midpoint: the centre point between locations
The geographic midpoint is the centre of gravity of the locations on the globe. Each latitude φ and longitude λ becomes a 3D unit vector x = cos φ cos λ, y = cos φ sin λ, z = sin φ; the vectors are averaged (weighted, if you like) and the result is turned back into latitude = atan2(z, √(x² + y²)) and longitude = atan2(y, x). Unlike a plain average of the numbers, this works across the 180° meridian and near the poles. For two places it is exactly the halfway point on the great circle: Riga and Tallinn meet at 58.1937° N, 24.4181° E.
Halfway point or meeting place: centre of minimum distance
If you want the place where the total travel distance for everyone is smallest, choose the centre of minimum distance. It is found by an iterative search that starts at the geographic midpoint and keeps moving towards lower Σ wᵢ·dᵢ until the step is below 1 cm. With weights (for example the number of people at each address) the point moves towards the heavier locations.
Simple average of latitude and longitude
Averaging the coordinates directly is quick and close enough for points a few kilometres apart, but it is wrong over long distances and fails across the antimeridian: 170° E and 170° W average to 0° instead of 180°. The calculator shows all three methods side by side, with the distance from the midpoint to each location as the crow flies.
Related tools
Measure a single leg with the distance between coordinates calculator, look up an address with the coordinates finder, or check the local time at the midpoint with the city time zone finder.
Formula
xᵢ = cos φᵢ · cos λᵢ, yᵢ = cos φᵢ · sin λᵢ, zᵢ = sin φᵢx̄ = Σ wᵢxᵢ / Σ wᵢ (same for ȳ, z̄)λ = atan2(ȳ, x̄), φ = atan2(z̄, √(x̄² + ȳ²))centre of minimum distance: minimise Σ wᵢ · dᵢ (great-circle dᵢ)simple average: φ̄ = Σ wᵢφᵢ / Σ wᵢ, λ̄ = Σ wᵢλᵢ / Σ wᵢ
How it is calculated
- Enter each location as an address, coordinates (DD, DMS, DDM) or with “Use my location”.
- Press “Add location” for more rows, or open the map and click to add places.
- Optionally tick “Use weights” and enter a weight per location, e.g. the number of people.
- Choose the method: geographic midpoint, centre of minimum distance or simple average.
- Read the midpoint in decimal degrees and DMS, find the nearest address and the distance to each location.
- Open “How it is calculated” to see the vectors and averages with your numbers, and copy the shareable link.
Questions
How is the geographic midpoint calculated?
Each location is turned into a 3D point on a unit sphere (x = cos φ cos λ, y = cos φ sin λ, z = sin φ). The x, y and z values are averaged, weighted if you set weights, and converted back: longitude = atan2(y, x), latitude = atan2(z, √(x² + y²)).
Which method should I use to find a halfway meeting point?
For two people both give the same halfway point. For three or more, the centre of minimum distance minimises the total distance everyone travels, while the geographic midpoint is the centre of gravity. Remember the distances are straight-line, so check the actual roads before you meet.
Why not just average the latitudes and longitudes?
Because the Earth is round. Lines of longitude converge towards the poles, and longitude jumps from +180° to −180°. Points at 170° E and 170° W would average to 0° (London's meridian) instead of 180°. The 3D method avoids both problems.
Can I add more than two locations or weight them?
Yes. There is no limit on locations: press “Add location” or click the map. Tick “Use weights” to give each place a weight, such as how many people start there or how often you visit it; the midpoint moves towards the heavier places. The shareable link stores all points and weights.