Quadrilateral Area From Coordinates Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The area of any quadrilateral is calculated directly from the coordinates of its four vertices, given in order, using the shoelace formula. For a square with vertices (0,0), (4,0), (4,4), and (0,4), the area is exactly 16.
Explanation
Unlike our area calculators for regular shapes such as the rectangle, the rhombus, or the parallelogram, which require knowing specific dimensions (sides, diagonals, angles), this formula works directly from the (x, y) coordinates of the four vertices, with no need to know any length or angle beforehand — useful whenever the shape is irregular, or its vertices are already known as coordinates (a cadastral plan, a technical drawing, GPS data). This method, called the shoelace formula (referring to the crossing pattern of the terms, reminiscent of lacing a shoe), works for any simple quadrilateral — convex or concave — provided the vertices are entered in order (clockwise or counterclockwise, either way) and the sides never cross. The result stays valid whatever the direction the vertices are entered in, thanks to the absolute value applied at the end of the calculation, which eliminates the negative sign that a reversed traversal direction would produce.
Example: rectangle with vertices (0,0), (5,0), (5,3), (0,3)
Inputs
Vertices: (0,0), (5,0), (5,3), (0,3).
Calculation
Area = ½ × |0×(0−3) + 5×(3−0) + 5×(3−0) + 0×(0−3)| = ½ × |0+15+15+0| = ½ × 30 = 15.
Result
The area of this quadrilateral is 15 square units.
Frequently asked questions
In what order should the four vertices be entered?
The vertices must be entered in the order they actually follow around the quadrilateral — clockwise or counterclockwise, either way, but never in a random order that would make the sides cross. Entering the vertices in the wrong order (for example by swapping two non-adjacent vertices) would give a bowtie shape rather than a simple quadrilateral, and an incorrect area result.
Does this formula work for a concave (non-convex) quadrilateral?
Yes, the shoelace formula remains valid for a concave quadrilateral (the one whose one interior angle exceeds 180°, giving it an arrow shape), as long as the vertices are entered in order and the sides do not cross. It does fail for a self-intersecting figure (sides crossing), which is no longer a simple quadrilateral in the geometric sense.
Why use an absolute value at the end of the formula?
Because the result of the calculation before the absolute value can be negative depending on the direction the vertices were entered (clockwise gives one sign, counterclockwise the other): this sign actually corresponds to a notion of signed area, useful in some advanced mathematical contexts, but of no interest here where you simply want the surface, always positive by definition.