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Coordinates

Geographic center calculator

Add several places (search for them, tap the map or paste a list) and see their center worked out three ways, weighted if you like. Or draw an area and see where its center of gravity is.

Add places
Paste a list of coordinates or a CSV

One place per line, in any format (decimal, degrees, UTM…), with or without a name. A number at the end is used as the weight.

Places on the list
#PlaceRemove
1New York40.6627° N, 73.9387° W
2Los Angeles34.0194° N, 118.4108° W
3Chicago41.8370° N, 87.6849° W
4Houston29.7857° N, 95.3888° W
5Phoenix33.5722° N, 112.0901° W

Example The five largest US cities. Add your own places and the example goes away.

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G · Center of gravity M · Least total distance S · Simple average

Result

Center of gravity

37.0553° N, 98.1429° W

Total distance
4,481 mi

The average of the places as points on the sphere, brought back to the surface. With two places, it’s the midpoint.

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Least total distance

35.6641° N, 97.6614° W

Total distance
4,461 mi
From the center of gravity
99.6 mi

The point with the smallest total distance to all of them: the best place to meet if everyone travels in a straight line.

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Simple average

35.9754° N, 97.5027° W

Total distance
4,462 mi
From the center of gravity
82.5 mi

Just the average of the latitudes and of the longitudes. Close to the others when the places are near each other; otherwise, way off.

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How to use it

  1. Add the places

    Search for towns or cities, tap the map, or paste a list of coordinates or a CSV. Give each one a weight if you want.

  2. Compare the three centers

    Center of gravity (G), least total distance (M) and simple average (S), each with its total distance to all the places.

  3. Or draw an area

    In “An area”, tap the map to mark its corners and you’ll get its center, area and perimeter.

What it’s for

  • Where to meet

    The least-total-distance point is the one that makes the group travel the least overall.

  • Where to put a warehouse or a store

    With customers weighted by orders or population, the center moves toward where the demand is.

  • The center of a country or a region

    Draw the area and see where its center of gravity falls, like the center of the lower 48.

  • See why the easy math fails

    Averaging latitudes and longitudes goes badly wrong with faraway places or on both sides of the date line.

How it’s calculated

There isn’t one single “center”: it depends on what you need. Mapsfera works out three on the sphere and plots them on the map, each with its total distance to the places, measured on the WGS84 ellipsoid.

The center of gravity turns each place into a 3D point (x, y, z), averages them (weighted, if you like) and projects the result back to the surface. The least-total-distance point, the geometric median, has no formula: it’s found step by step with Weiszfeld’s method, which at each step moves closer to the nearby places. The simple average just averages the coordinate numbers.

The center of an area is its center of gravity: that of the whole surface, not of its corners. It uses the exact formula for the moment of a polygon on the sphere.

Center of gravity

v = Σ wᵢ·(cos φᵢ cos λᵢ, cos φᵢ sin λᵢ, sin φᵢ); center = v / |v|

φ is latitude, λ longitude and w the weight of each place.

Least total distance

minimum of Σ wᵢ·d(x, placeᵢ)

d is the distance over the Earth. Weiszfeld step: the new position is the average of the directions to each place, each weighted by wᵢ / dᵢ.

Example: New York, Los Angeles, Chicago, Houston, and Phoenix
Center of gravity (G) 37.0553° N, 98.1429° W · total 4,481 mi (7,211 km)
Least total distance (M) 35.6641° N, 97.6614° W · total 4,461 mi (7,179 km)
Simple average (S) 35.9754° N, 97.5027° W · total 4,462 mi (7,181 km)
Distance from the simple average to the center of gravity 82.5 mi (133 km)

With faraway places, the simple average really breaks. Between Suva, Fiji and Apia, Samoa, on either side of the International Date Line, the center of gravity falls at 16.0° S, 176.6° W, in the middle of the Pacific, but averaging the longitudes gives 16.0° S, 3.3° E: on the other side of the world.

The geographic center of the United States

The U.S. Geological Survey lists the geographic center of the 50 states at 44° 58′ N, 103° 46′ W, west of Castle Rock in Butte County, South Dakota, and that of the 48 contiguous states at 39° 50′ N, 98° 35′ W, near Lebanon in Smith County, Kansas. The USGS itself points out that there’s no generally accepted definition of a geographic center and that these positions are approximations.

With the Census outlines this site uses, the center of gravity of the lower 48 comes out at 39.7583° N, 99.0088° W, in Smith County, Kansas, 23.2 mi (37.4 km) from the USGS point; for all 50 states, at 44.7617° N, 103.9838° W, in Butte County, South Dakota, 17.7 mi (28.5 km) away.

The center of Spain

Tradition puts the center of the Iberian Peninsula at the Cerro de los Ángeles hill in Getafe, just south of Madrid (40° 18′ 31″ N, 3° 41′ 04″ W); the Madrid region’s tourism site calls it “traditionally considered the geographic center of the Iberian Peninsula”. With the site’s outlines, the center of gravity of mainland Spain (IGN provinces) is at 40.3623° N, 3.5644° W, in the municipality of Madrid (Madrid province), 7.34 mi (11.8 km) from the hill, and that of the whole peninsula, Portugal included (Natural Earth), at 40.2675° N, 4.2401° W, in the municipality of Santa Cruz del Retamar (Toledo province), 29.5 mi (47.5 km) away.

Frequently asked questions

Which center should I use?

To meet up with several people or pick a location, the least-total-distance point: it minimizes everyone’s combined travel. The center of gravity is the “midpoint” of several places and shifts more when you add a faraway one. The simple average only works when the places are close together and away from the poles and the date line.

What are weights for?

To make some places count more than others: if one city has ten times the population of another, using population as the weight pulls the center toward it. Type the weights in the list or paste them at the end of each CSV line.

Why can the center of an area fall outside it?

Because it’s the balance point of the whole surface. In a C-shaped, horseshoe or crescent area that point lies in the gap, just like the center of gravity of a ring is in thin air.

Does it account for the Earth being round?

Yes: all three centers and the area center are worked out on the sphere, and distances on the WGS84 ellipsoid. That’s why it works with places anywhere in the world, at the poles or on both sides of the date line.

Related tools

All tools
Latitude and longitude finder Search for any place or address, or tap the map, and copy its latitude and longitude in one click. Plot coordinates on a map Paste a latitude and longitude in any format and see the exact spot on the map, ready to share. Coordinate converter Convert coordinates between decimal degrees, DMS, UTM, MGRS, geohash and Plus Codes in one go. UTM to latitude and longitude Turn a UTM zone, easting and northing into latitude and longitude, and see the point on the map.

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