Polygon Area (lat/lng)
Compute the area of a polygon on Earth from lat/lng vertices.
Overview
The Polygon Area tool computes the area of a closed polygon defined by latitude/longitude vertices. Paste in a list of corners, in order around the perimeter, and the tool returns the enclosed area in square metres, square kilometres, hectares and acres. It is the geographic answer to "how big is this plot of land?" or "what is the area of this country, lake, or national park?"
Computing area on a sphere is meaningfully different from computing it on a flat plane. A naive shoelace formula on raw lat/lng values returns a number in "square degrees" that is nearly meaningless, because longitude degrees shrink as you move toward the poles. This tool projects the polygon onto the WGS84 sphere and applies a spherical polygon area formula so the result is correct anywhere on Earth - whether the polygon is a small back-garden plot, a city boundary or a continent-sized region.
How it works
The tool uses the spherical excess formula derived from Green's theorem on a sphere. For a polygon with vertices (φ₁, λ₁) ... (φₙ, λₙ) in radians, the signed area is R² · Σ (λ_{i+1} - λ_{i-1}) · sin(φ_i) / 2, summed over all vertices with indices taken modulo n, where R is the radius of Earth - 6371 km. Taking the absolute value gives the unsigned area. The sign of the raw sum indicates winding order, which is useful for detecting whether the polygon is given clockwise or counter-clockwise.
The polygon is assumed to be simple (no self-intersections) and lying on one side of the sphere - the tool does not handle polygons larger than a hemisphere. Holes are not supported in the basic form; subtract the inner polygon's area from the outer polygon's area to get a net value.
Examples
- A 1 km square at the equator - returns roughly 1.00 km² (or 100 ha, 247 acres).
- The contiguous United States outline - returns roughly 8.1 million km², matching published values within rounding.
- A small rectangular parking lot, 50 m by 30 m - returns 1,500 m² or 0.15 ha.
- A polygon that wraps the wrong way around a continent - the absolute area looks wrong because the formula computes the complement; reverse the vertex order to fix it.
FAQ
Do I need to close the polygon?
The tool treats the polygon as closed automatically - the last vertex connects back to the first. You can supply the closing vertex if you wish, but it is not required.
What about polygon holes?
The basic tool computes one polygon at a time. For shapes with holes (an outer ring and one or more inner rings), compute each ring's area separately and subtract the inner from the outer.
Why does my country differ from the published value?
Published country areas often include or exclude inland water, overseas territories or disputed regions in different ways. Small differences are usually data-source choices rather than formula error.
Does the formula work at the poles?
Yes, including over the poles - the spherical excess formula is well-defined everywhere on the sphere. The only restriction is that the polygon should not cover more than a hemisphere.