The Area by Coordinates Formula is a mathematical method used to calculate the area of a polygon when the coordinates of its vertices are known.
You can use:
It is the fastest and most reliable method for calculating the area of:
This formula is widely known in surveying as the Coordinate Method, and is used in total station processing, GIS software, CAD programs, and online calculators.
Let a polygon have n vertices with coordinates: (x1,y1),(x2,y2),…,(xn,yn)(x_1, y_1), (x_2, y_2), …, (x_n, y_n)(x1,y1),(x2,y2),…,(xn,yn)
The area AAA is: A=12∣∑i=1n−1(xiyi+1−yixi+1)+(xny1−ynx1)∣A = \frac{1}{2} \left| \sum_{i=1}^{n-1}(x_i y_{i+1} – y_i x_{i+1}) + (x_n y_1 – y_n x_1) \right|A=21i=1∑n−1(xiyi+1−yixi+1)+(xny1−ynx1)
This method works for any polygon, including irregular shapes.
Surveyors love this formula because:
It works perfectly for both small and large areas.
Straight boundaries, irregular parcels, concave shapes — no problem.
Total stations, GPS devices, GIS systems, and calculators all use this internally.
Even a 20-point polygon can be calculated in seconds.
As long as coordinates are in meters, the result is in square meters.
| Point | X (Easting) | Y (Northing) |
|---|---|---|
| 1 | 100 | 200 |
| 2 | 140 | 250 |
| 3 | 180 | 220 |
| 4 | 130 | 180 |
Sum A = 25,000 + 30,800 + 32,400 + 26,000 = 114,200
Sum B = 28,000 + 45,000 + 28,600 + 18,000 = 119,600
114,200−119,600=−5,400114,200 – 119,600 = -5,400114,200−119,600=−5,400
∣−5,400∣=5,400| -5,400 | = 5,400∣−5,400∣=5,400
Area=5,4002=2,700 sq. meters\text{Area} = \frac{5,400}{2} = 2,700 \text{ sq. meters}Area=25,400=2,700 sq. meters
You can convert this:
| Unit | Conversion |
|---|---|
| 1 sq meter | 10.7639 sq ft |
| 1 hectare | 10,000 sq m |
| 1 acre | 4,046.856 sq m |
| 1 sq km | 1,000,000 sq m |
You cannot directly apply the formula using degrees.
You must first convert:
Then apply the area formula in meters.
This is why most online calculators automatically convert geodetic coordinates to planar coordinates.
Used to calculate land boundaries and plot areas.
Used in earthwork, site layout, and planning.
Farmers measure field areas for irrigation, pesticide planning, yield estimation.
Used for calculating polygon areas in spatial analysis.
Photogrammetry software uses polygon coordinates to compute field areas.
Points must be in:
Random order gives wrong results.
The last point is NOT repeated — the formula closes automatically.
Degrees will give meaningless area values.
Meters with feet causes major errors.
The formula works for concave polygons, but self-intersecting (bow-tie) shapes fail.
People search for Area by Coordinates Calculator because:
Manual calculation is slow for large datasets.
Avoids human multiplication errors.
UTM, lat/long, site grid, CSV files.
Surveyors use mobile calculators during site visits.
Below is a powerful FAQ block designed to boost ranking.
It is a mathematical formula (Shoelace Formula) used to calculate the area of any polygon using its coordinates (x, y).
Yes, but only after converting lat/long to UTM or another projected coordinate system.
Yes. Points must be arranged clockwise or counter-clockwise. Random order produces incorrect results.
If coordinates are in:
Yes. It is highly accurate and used in professional surveying.
Yes. It works for any polygon, no matter how irregular.
Use an online calculator or GIS software. Manual computation is not practical for large datasets.
Because the pattern of multiplication resembles lacing a shoe.
No. The formula only uses X and Y (2D area).
For 3D surface area, a different method is used.
Perimeter uses distances between points.
Area uses cross-multiplication of coordinates.
The Area by Coordinates Formula is one of the most important tools in surveying, GIS, engineering, land mapping, and agriculture. It is fast, reliable, and easy to automate. Whether you’re calculating the area of a farm, plot, building footprint, or drone-mapped polygon, this method delivers accurate results every time.