Area of Quadrilateral Calculator
Choose your quadrilateral type or method, enter the measurements, and get the area instantly with a step-by-step breakdown. Covers rectangles, squares, parallelograms, rhombuses, kites, trapezoids, coordinate polygons, and general irregular quadrilaterals via Bretschneider's formula. Switch between metric and imperial units at any time.
What is a quadrilateral?
A quadrilateral is any polygon with exactly four straight sides and four vertices. The family includes well-known regular shapes - rectangles, squares, parallelograms, rhombuses, kites and trapezoids - as well as completely irregular four-sided figures. Because the sides and angles can vary so widely, several different formulas exist, each suited to a different set of known measurements. This calculator covers all of the common cases so you can pick the method that matches the data you already have.
Which formula should I use?
Use "Rectangle" when you know the two side lengths of a rectangle (base and height). Use "Square" when you know the single side length. Use "Parallelogram" when you know the base and the perpendicular height (not the slant height). Use "Rhombus or Kite" when you have both diagonals: area is half the product of the diagonals for both shapes. Use "Trapezoid" when you know both parallel sides and the perpendicular height between them. Use "General - two diagonals + angle" when you know only the lengths of the diagonals and the angle at which they cross: this works for any quadrilateral. Use "General - four sides + two opposite angles" (Bretschneider's formula) when you have all four side lengths plus two opposite interior angles. Use "General - four vertex coordinates" (the shoelace formula) when you have the x and y coordinates of all four corners, for example from a map or a CAD drawing.
Bretschneider's formula explained
Bretschneider's formula is the most general area formula for a quadrilateral given only side lengths and angles. With semi-perimeter s = (a+b+c+d)/2 and two opposite angles alpha and gamma, the area is the square root of (s-a)(s-b)(s-c)(s-d) minus a*b*c*d times cos-squared of half the sum of alpha and gamma. When alpha plus gamma equals 180 degrees the quadrilateral is cyclic (inscribed in a circle), the cosine term disappears, and the formula reduces to Brahmagupta's simpler version. If all four sides are equal and all angles are right angles you recover the square formula. The formula requires that the four sides actually form a valid quadrilateral, so the inner expression under the square root must be non-negative.
The shoelace formula for coordinate-defined shapes
When you know the coordinates of every vertex, the shoelace (Gauss) formula gives the exact area regardless of the shape's orientation or symmetry. List the vertices in order (either all clockwise or all counterclockwise), form the sum of (x_i * y_{i+1} - x_{i+1} * y_i) for each consecutive pair of vertices (treating the last vertex as adjacent to the first), then take half the absolute value of that sum. This formula generalises beyond quadrilaterals to any simple polygon. It is particularly useful when working with GPS coordinates, architectural plans, or survey data where you already have numerical coordinates.
Quadrilateral area formulas at a glance
| Shape | Formula | Notes |
|---|---|---|
| Rectangle | A = a x b | Any two adjacent sides |
| Square | A = a^2 | All sides equal |
| Parallelogram | A = b x h | h is the perpendicular height |
| Rhombus | A = (d1 x d2) / 2 | d1 and d2 are the two diagonals |
| Kite | A = (d1 x d2) / 2 | Same formula as rhombus |
| Trapezoid | A = ((a + b) / 2) x h | a and b are the two parallel sides |
| General (diagonals) | A = 0.5 x p x q x sin(theta) | theta = angle between diagonals |
| General (Bretschneider) | A = sqrt((s-a)(s-b)(s-c)(s-d) - abcd*cos^2((alpha+gamma)/2)) | General 4-sided polygon |
| General (coordinates) | A = |sum of cross products| / 2 | Shoelace / Gauss formula |
Standard formulas for common quadrilateral types. Variables: a/b = sides, h = perpendicular height, d1/d2 = diagonals, s = semi-perimeter.
Frequently asked questions
What is the area formula for a general quadrilateral?
There is no single formula that works from side lengths alone for a general quadrilateral, because unlike a triangle it is not rigidly determined by its four sides. You always need at least one additional measurement: an angle, a diagonal, or the vertex coordinates. The most common general formulas are: area = 0.5 * p * q * sin(theta) using the two diagonals and the angle between them; Bretschneider's formula using all four sides plus two opposite angles; and the shoelace formula using the x,y coordinates of all four vertices.
Does the order of vertices matter for the coordinate method?
Yes. Enter the four vertices in order around the perimeter, either all clockwise or all counterclockwise. If you list them out of order the shoelace formula treats the shape as a self-intersecting (bow-tie) polygon and gives the wrong area. A simple check: if your result looks too small, try reordering the vertices.
What is the difference between a rhombus and a kite?
A rhombus has all four sides equal and its diagonals bisect each other at right angles. A kite has two pairs of adjacent equal sides and its diagonals are perpendicular but only one diagonal bisects the other. Despite those geometric differences, the area formula is the same for both: half the product of the two diagonals.
Can I use this calculator for a cyclic quadrilateral?
Yes. Select the Bretschneider method and enter all four side lengths and two opposite angles. For a cyclic quadrilateral inscribed in a circle, the two opposite angles always sum to 180 degrees, which simplifies the formula to Brahmagupta's version. The calculator handles both cases automatically.
Why does the parallelogram formula use perpendicular height, not the slant side?
The area of a parallelogram equals base multiplied by the perpendicular height, that is the shortest distance between the two parallel bases. The slant side is longer than that perpendicular distance, so using the slant side would overestimate the area. If you only know the base and the slant side, you also need the interior angle: area = base * slant * sin(angle).