What this calculator does
A quadrilateral is any four-sided flat shape, and unlike a square, rectangle or specific type of polygon, its four sides and angles do not need to follow any particular rule. That generality makes a single fixed formula, like side times side, impossible; instead, the area of an arbitrary quadrilateral is worked out from the coordinates of its four corners.
This calculator uses the shoelace formula, a standard method from coordinate geometry that finds the area of any polygon, including an irregular quadrilateral, directly from the (x, y) position of each corner, entered in order around the shape.
The formula
Enter the four corners in order as you would trace around the shape, either clockwise or anticlockwise, but not jumping across it. The shoelace formula multiplies each corner's x-coordinate by the difference of its two neighbouring y-coordinates, sums those across all four corners, and takes half the absolute value of the total.
| Term | Meaning |
|---|---|
| (x, y) | The coordinates of each of the four corners, in any consistent unit. |
| Shoelace formula | A coordinate-geometry method that computes the area of a polygon from its vertex coordinates, named for the crisscross pattern the multiplications trace out. |
The inputs explained
| Field | What to enter |
|---|---|
| Corner 1, x | The x-coordinate of the first corner. |
| Corner 1, y | The y-coordinate of the first corner. |
| Corner 2, x | The x-coordinate of the second corner, going around the shape from the first. |
| Corner 2, y | The y-coordinate of the second corner. |
| Corner 3, x | The x-coordinate of the third corner. |
| Corner 3, y | The y-coordinate of the third corner. |
| Corner 4, x | The x-coordinate of the fourth corner. |
| Corner 4, y | The y-coordinate of the fourth corner. |
When to use it
An irregular block of land or floor plan
A parcel of land or a room that is not a clean rectangle, but has four straight boundary lines at surveyed or measured coordinates, has its area worked out exactly this way rather than by guessing at an average width.
A maths problem given as coordinates
When a quadrilateral is defined by four (x, y) points rather than by side lengths and angles, the shoelace formula is the direct route to its area without needing to first work out what specific type of quadrilateral it is.
Checking a CAD or survey area figure
Recomputing the area from the same four corner coordinates independently checks a figure produced by drafting software or a land survey.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How area changes as one corner moves outward
Three corners held fixed at (0, 0), (6, 0) and (1, 4), with the third corner's x-coordinate moved outward while its y stays at 5.
| Third corner, x | Area | Perimeter |
|---|---|---|
| 6 | 24.500 | 20.222 |
| 8 | 28.500 | 22.579 |
| 10 | 32.500 | 25.582 |
| 12 | 36.500 | 28.979 |
| 15 | 42.500 | 34.454 |
| 20 | 52.500 | 44.015 |
Questions
What order should the corners be entered in?
Enter them in order as you trace around the perimeter of the shape, either clockwise or anticlockwise, without skipping across the middle. Entering them out of order, so the sides cross over each other, gives an incorrect area.
Does it matter whether I go clockwise or anticlockwise?
No. The shoelace formula takes the absolute value of the result, so either direction around the shape gives the same positive area.
Can this handle a concave quadrilateral, where one corner points inward?
Yes, as long as the four corners are still entered in order around the boundary without the sides crossing. The shoelace formula correctly handles concave as well as convex quadrilaterals.
Why not just use a formula for a specific shape like a rectangle or kite?
A rectangle, kite, trapezium or parallelogram each has its own simpler formula because it has a known regularity, such as equal sides or parallel sides. A general quadrilateral has no such guarantee, so its area has to come from the actual corner positions instead.
For a quadrilateral with two pairs of parallel sides, see the area of trapezium calculator. For a four-sided shape defined by its two diagonals instead, see the area of a kite calculator.