Irregular Polygon Area Calculator
Calculate the area of an irregular polygon from the X and Y coordinates of its corner points.
Enter the vertices of the polygon in order around its boundary. The calculator uses the shoelace formula to calculate irregular polygon area and reports the result in your selected square units.
Calculate irregular polygon area
Enter at least three coordinate points. Work clockwise or counter-clockwise and keep the points in order around the outside of the shape.
Polygon coordinates
Enter each corner point in boundary order.
Polygon area
- Area
- 32.00 m²
- Square metres
- 32.00 m²
- Square feet
- 344.45 ft²
- Number of vertices
- 4
- Calculation method
- Shoelace formula
Shape preview
The preview shows the order in which the coordinate points are connected.
Shoelace calculation
Area = ½ × |Σ(xᵢyᵢ₊₁ − yᵢxᵢ₊₁)|
Area = 32.00 m²
How to enter irregular polygon coordinates
Start at any corner of the polygon and enter each vertex consecutively around the outside boundary. You can travel clockwise or counter-clockwise, but do not jump across the polygon from one side to another.
The points should describe a simple, non-self-intersecting polygon. If the connecting lines cross each other, the shoelace result may not represent the physical area you intended to measure.
How to calculate the area of an irregular polygon
- Choose the measurement unit used by your coordinates.
- Enter the X and Y coordinates for at least three vertices.
- Add more points when the polygon has additional corners.
- Keep the points in clockwise or counter-clockwise order around the boundary.
- Select Calculate area.
The calculator connects the last point back to the first automatically, so you do not need to enter the starting coordinate twice.
Irregular polygon area formula
When the coordinates of all polygon vertices are known, the area can be calculated with the shoelace formula. It is also called the surveyor's formula or Gauss's area formula.
Each X coordinate is multiplied by the Y coordinate of the following point. The reverse products are then subtracted, all differences are added together and the absolute value is divided by two.
The final vertex is paired with the first vertex to close the polygon.
Irregular polygon area example
Consider a four-sided polygon with these coordinates:
Applying the shoelace formula gives:
½ × |(0×0 + 8×4 + 8×4 + 0×0) − (0×8 + 0×8 + 4×0 + 4×0)|
Area = 32 square unitsArea of an irregular pentagon
An irregular pentagon has five sides that do not necessarily have equal lengths or equal angles. If the coordinates of its five vertices are known, its area can be calculated in exactly the same way as any other irregular polygon.
Enter the five corner coordinates as points A through E in boundary order. The calculator then applies the shoelace formula to all five vertices and closes the shape from point E back to point A.
This allows the same calculator to be used as an irregular pentagon area calculator without requiring a separate calculation method.
What can an irregular polygon area calculation be used for?
Coordinate-based polygon area calculations can be useful for irregular plots, floor plans, drawings, geometric problems, landscape areas and other shapes where simple length × width formulas do not apply.
The result represents a flat two-dimensional area. It does not account for ground slope, elevation changes or curved boundaries between the coordinate points.
Can polygon coordinates be negative?
Yes. X and Y values can be positive, negative or zero. The polygon does not need to begin at the origin and can be located anywhere on the coordinate plane.
Moving the entire polygon to a different position on the coordinate plane does not change its area.
Important calculation limitations
This calculator treats every edge between adjacent coordinates as a straight line.
For curved boundaries, the result will depend on how closely the entered points approximate the curve.
The calculator is intended for simple polygons whose boundary does not cross itself.
Frequently asked questions
What is an irregular polygon?
An irregular polygon is a polygon whose sides or angles are not all equal. Triangles, quadrilaterals, pentagons and polygons with many more vertices can all be irregular.
How do you calculate irregular polygon area?
If the X and Y coordinates of the vertices are known, the shoelace formula can calculate the enclosed area. The vertices must be entered in order around the polygon.
Does the first point have to be 0,0?
No. The first coordinate can be anywhere. The position of the polygon on the coordinate plane does not affect its area.
Can I calculate the area of an irregular pentagon?
Yes. Add five coordinate points corresponding to the five corners of the pentagon and enter them in clockwise or counter-clockwise order.
Do I need to repeat the first coordinate at the end?
No. The calculator automatically joins the final vertex back to the first vertex when applying the shoelace formula.
Can I use feet instead of metres?
Yes. Choose feet from the unit selector. You can also enter coordinates in metres, centimetres, inches or yards.
Can I enter negative coordinates?
Yes. Negative X and Y coordinates are supported.