Polygon & Multi-Point Centroid Calculator

The centroid is the geometric centre of a shape — the arithmetic average of all the points that make it up, weighted by the area or length they contribute. For a physical object of uniform density, the centroid coincides with the centre of mass, which is why engineers sometimes use the terms interchangeably.

In GIS and spatial analysis, the centroid serves as a representative point for a polygon feature. QGIS and ArcGIS use the polygon centroid as the default anchor for feature labels on a map. It is also used in logistics (finding a central depot location), remote sensing (referencing a detected landslide or flood extent), and urban planning (locating the point of minimum aggregate travel distance within a service zone).

Importantly, the centroid of a polygon is not simply the centre of the bounding box. It is the area-weighted geometric average of the polygon vertices. For a long, narrow country like Chile, the centroid sits far south of the bounding-box midpoint because the wider southern region contributes more area to the average.

This calculator uses the shoelace formula (Gauss's area formula) to compute the signed area and area-weighted centroid of a polygon simultaneously. For a simple polygon with vertices (x₁,y₁) … (xₙ,yₙ):

A  = ½ × Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ) Cx = (1 / 6A) × Σ (xᵢ + xᵢ₊₁)(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ) Cy = (1 / 6A) × Σ (yᵢ + yᵢ₊₁)(xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)

The formula is exact for any simple (non-self-intersecting) polygon regardless of the number of sides. Polygons with holes (donuts) are handled by subtracting the hole centroids weighted by their areas from the outer ring centroid.

For multi-polygons, each constituent polygon's centroid is weighted by its own area and combined into a single result. For line geometries, the centroid is the length-weighted midpoint of the segments.

A multi-point centroid (the centre of gravity of a point cloud) is the arithmetic mean of all coordinates:

Cx = (x₁ + x₂ + … + xₙ) / n Cy = (y₁ + y₂ + … + yₙ) / n

Every point is weighted equally. This is used in logistics problems (finding an optimal depot location for n delivery addresses) and as the update step in k-means clustering. Use this mode when you have a set of discrete locations — customer addresses, GPS waypoints, survey control points — and want their geographic centre.

A polygon centroid weights each part of the polygon by the area it contributes, not just the vertices. For regular shapes like squares and equilateral triangles, both methods give the same result. For irregular or elongated polygons, only the area-weighted centroid correctly reflects the true geometric centre.

• GIS label placement — QGIS and ArcGIS use the polygon centroid as the default anchor for feature labels. If the automatic centroid falls outside an irregular polygon (e.g. a concave bay or an atoll), use this tool to compute a more intuitive placement coordinate.
• Logistics and supply-chain — Given a set of delivery addresses or store locations, the multi-point centroid provides the initial estimate for an optimal depot or distribution-centre location before further route-optimisation.
• Urban and land-use planning — The centroid of a service-area polygon (a school catchment, a flood-risk zone, a rezoning boundary) identifies the point that minimises aggregate travel distance within the zone.
• Remote sensing and disaster response — The centroid of a detected landslide scar, wildfire perimeter, or flood extent is used as the reference coordinate in event reports and in spatial queries against infrastructure databases.
• Game development and simulation — The centroid of a spawn zone or region of interest is used as the default spawn point, camera anchor, or minimap icon position.

This calculator uses a planar (Euclidean) centroid — it treats longitude and latitude as X and Y coordinates on a flat grid. For areas smaller than a few hundred kilometres, the planar centroid is accurate to within a few metres and is perfectly suitable for city, district, and country-scale work.

For very large polygons spanning many degrees of longitude at high latitudes, the planar centroid can deviate from the true geodesic centroid. This is because one degree of longitude is only 55.8 km wide at 60°N versus 111.3 km at the equator — a planar calculation treats them as equal, overstating the influence of eastern and western extents at high latitudes.

As a rule of thumb: for polygons smaller than a single country, the planar centroid error is below 1 km and is negligible for mapping purposes. For continental or ocean-basin scale analysis, project to an equal-area CRS (such as EPSG:6933 WGS84 / NSIDC EASE-Grid 2.0 Global) before computing the centroid.