How to use it
- Type one corner per line in Corners, one per line, as latitude and longitude or in another form the coordinate converter reads.
- Keep the corners in order around the outline, so that the sides do not cross.
- Read Area in square metres, and Hectares, Dönüm (1 000 m²) or Acres for the unit you need.
- Open Working to see the spherical excess and the area step by step.
How the area is found
On a sphere, a polygon whose sides are great-circle arcs has angles that add up to more than the flat total. That excess, the spherical excess E, is the area divided by the square of the radius. The tool splits the polygon into triangles from its first corner, finds the excess of each triangle with the Van Oosterom and Strackee formula, adds them up and multiplies by R².
E = Σ 2 × atan2(a · (b × c), 1 + a·b + b·c + c·a), over the triangles (a, bᵢ, bᵢ₊₁)
Area = |E| × R², with R = 6 371 008.8 m, the mean radius
Perimeter = Σ 2R × atan2(√h, √(1 − h)), where h is the haversine of each side- The whole sphere has an excess of 4π steradians, so a polygon's excess in steradians times R² gives its area.
- The unit vectors a, b and c are the corners on the sphere: a is the first corner, and b and c the next two.
Accuracy on the real Earth
The Earth is an ellipsoid, flattened at the poles. The sphere of mean radius gives areas within about 0.1 % of the WGS 84 value near Ankara, and within about 0.7 % for one-degree squares from the equator to 70° north (compared with the area element of the ellipsoid). The area is also taken at sea level: the real surface of a slope is larger than its projection.
- 1 dönüm is 1 000 m², the figure the Turkish Wikipedia article on the dönüm gives for the unit in use today. The older dönüm, used before the 1945 law, was 918.393 m².
What this tool does not cover
- Polygons whose sides cross, such as a bow tie: their area has no meaning.
- Legal boundaries. A land registry area comes from the cadastre and the official survey; use that for legal purposes.
- Height and slope. The result is the area of the projection at sea level.
Worked example: a block near Ankara
The four corners 39.9334, 32.8597; 39.9334, 32.8697; 39.9244, 32.8697 and 39.9244, 32.8597 make a block 0.0090° of latitude (about 1.00 km) by 0.0100° of longitude (about 0.85 km at 39.93° N).
The sphere gives 853 334 m², which is 85.3334 hectares, 853.33 dönüm or 210.86 acres. The perimeter is 3 707 m.
| Result | Value |
|---|---|
| Area | 853 334 m² |
| Hectares | 85.3334 ha |
| Dönüm (1 000 m²) | 853.33 dönüm |
| Acres | 210.86 ac |
| Perimeter | 3 707 m |
Frequently asked questions
Why is the area on a sphere not the same as the area on a flat map?
On the curved surface the sides of a polygon are great-circle arcs, and the sum of its angles is more than the flat total. For a field of a few hectares the difference from a flat map is small; for large areas it is not.
What is a dönüm?
A dönüm is the Turkish unit of land: 1 000 square metres, which is 0.1 hectare. The older dönüm, used before the 1945 law, was 918.393 square metres.
How many dönüm are in a hectare and in an acre?
A hectare is 10 000 m², which is 10 dönüm. An international acre is 4 046.8564224 m², which is about 4.047 dönüm.
Is this the area of a registered parcel?
No. A registered parcel area comes from the land registry and the cadastral survey. This tool gives the area of the corners you type, on a sphere; use the official deed or survey for legal purposes.
Can I write the corners in degrees, minutes and seconds?
Yes. Each line can be decimal degrees, degrees minutes seconds, degrees and decimal minutes, UTM or MGRS, as the coordinate converter reads them. Blank lines are skipped, and a corner repeated at the end closes the polygon.
Why does a polygon whose sides cross give a wrong area?
The area is a signed sum taken in the order you type the corners. Where the sides cross, the parts traversed the other way are subtracted, so the result is not the area of the shape you drew. Order the corners around the outline.
Sources
- Movable Type Scripts: the haversine distance and the great-circle formulas on the sphere
- Wikipedia: Spherical excess (the area of a spherical polygon and its triangles)
- Wikipedia (Turkish): Dönüm (1 000 m², the unit in use today)
- NIST Special Publication 811, Appendix B: the acre and the other conversion factors
- Wikipedia: World Geodetic System (the WGS 84 ellipsoid)
Figures and text last checked on Tuesday, November 10, 2026.