For a simple polygon with ordered vertices (x₁,y₁), …, (xₙ,yₙ), its area is A = ½ |Σ(xᵢyᵢ₊₁ − yᵢxᵢ₊₁)|, connecting the last vertex back to the first. This is called the shoelace formula.
What the input must look like
Each point is a pair of numbers in the same flat coordinate system, such as meters east and meters north from an arbitrary origin. List points consecutively around the outer boundary—clockwise or counterclockwise. The list should trace a simple polygon, not jump across the interior.
Worked example with five vertices
Use points A(0,0), B(6,0), C(6,3), D(3,5), E(0,3). Append A to the end of the calculation to close the polygon.
| From → to | xᵢ × yᵢ₊₁ | yᵢ × xᵢ₊₁ |
|---|---|---|
| A → B | 0 × 0 = 0 | 0 × 6 = 0 |
| B → C | 6 × 3 = 18 | 0 × 6 = 0 |
| C → D | 6 × 5 = 30 | 3 × 3 = 9 |
| D → E | 3 × 3 = 9 | 5 × 0 = 0 |
| E → A | 0 × 0 = 0 | 3 × 0 = 0 |
| Total | 57 | 9 |
A = ½ × |57 − 9|
A = 24 square coordinate units
Verify the answer by geometry
The polygon fits inside a 6 × 5 bounding rectangle with area 30. It excludes two upper corner triangles, each with base 3 and height 2, so each has area 3. Thus 30 − 3 − 3 = 24, matching the shoelace result. A second method is valuable because one transposed digit can spoil a coordinate calculation.
What can go wrong?
- Wrong point order: jumping between vertices may create crossing edges and invalidate the intended shape.
- Forgetting the return pair: include the last-to-first contribution.
- Dropping the absolute value: clockwise and counterclockwise lists give opposite signed values; geometric area is nonnegative.
- Unclear units: if the coordinate units are meters, the area is m². If they are arbitrary drawing units, label the result accordingly.
Can I use GPS latitude and longitude?
Not directly for a reliable ground area. Latitude and longitude are angular measurements on a curved Earth; the flat shoelace formula treats them as Cartesian distances. For land measurements, use an appropriate projected coordinate system or geodesic area computation, and rely on professional survey data for legal boundaries.
The homepage calculator supports standard shapes rather than arbitrary vertex lists. For manual decomposition methods, read how to calculate irregular shapes.
Further mathematical reading: Wolfram MathWorld: Polygon Area.