Area of Quadrilateral Calculator
This calculator works out the area of any four-sided shape, covering rectangles and squares, parallelograms, trapezoids (trapeziums), rhombuses, kites, and general irregular quadrilaterals, using the formula that matches the shape you select. Start by choosing your quadrilateral type from the dropdown, and the input fields update to match: for a rectangle or square you enter the length and width; for a parallelogram, the base and the perpendicular height, not the slant side; for a trapezoid, the two parallel sides and the height between them; for a rhombus or kite, the lengths of the two diagonals; and for a general quadrilateral, the x and y coordinates of all four corners, entered in order around the shape. As you type, the calculator returns the area in square units, the shape type, and the formula used, shown in a results bar alongside a shape properties box and a step-by-step working panel that shows how each figure was substituted into the formula, so you can check your workings or follow the method for homework or site measurements. A worked example using default figures is included so you can see the process before entering your own numbers. For the general quadrilateral option using the shoelace formula, list the four vertices consistently, either clockwise or anti-clockwise, and make sure the shape does not cross over itself, or the result will be wrong.
1. Shape Type
2. Dimensions
Step-by-Step Working
Shape Properties
Worked Example
Quadrilateral Area Formulas
A quadrilateral is any polygon with exactly four sides and four angles. There are several common types, each with its own area formula.
| Shape | Formula | Notes |
|---|---|---|
| Rectangle / Square | A = length x width | All angles are 90 degrees. A square is a rectangle with equal sides. |
| Parallelogram | A = base x height | Height is the perpendicular distance between the two parallel bases, not the slant side length. |
| Trapezoid (Trapezium) | A = 0.5 x (a + b) x h | a and b are the two parallel sides; h is the perpendicular height between them. |
| Rhombus / Kite | A = 0.5 x d1 x d2 | d1 and d2 are the lengths of the two diagonals. For a rhombus, diagonals bisect each other at right angles. For a kite, one diagonal bisects the other. |
| General quadrilateral | Shoelace formula | Uses the (x, y) coordinates of all four vertices. Works for any simple (non-self-intersecting) quadrilateral. |
The Shoelace Formula for a General Quadrilateral
When a quadrilateral does not fit a standard type, the shoelace formula (also called the surveyor's formula) gives the exact area using the coordinates of its four vertices listed in order:
A = 0.5 x |(x1 x y2 - x2 x y1) + (x2 x y3 - x3 x y2) + (x3 x y4 - x4 x y3) + (x4 x y1 - x1 x y4)|
The vertices must be listed either all clockwise or all anti-clockwise around the shape. If they are listed in a mixed order, the formula will give an incorrect result. The absolute value ensures a positive area regardless of the direction of listing.
Trapezoid vs Trapezium: Which Name is Correct?
In New Zealand (following British usage), a four-sided shape with one pair of parallel sides is called a trapezium. In the United States, the same shape is called a trapezoid. This calculator uses both terms. The formula is the same: A = 0.5 x (a + b) x h, where a and b are the two parallel sides and h is the perpendicular height between them.
Parallelogram Height vs Side Length
A common mistake when calculating the area of a parallelogram is using the slant side length instead of the perpendicular height. The perpendicular height is the shortest distance between the two parallel bases, measured at a right angle to those bases. If you only know the side lengths and an interior angle, you can find the height using: h = side x sin(angle).
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Sources and method: Area formulas from standard Euclidean geometry. Shoelace (surveyor's) formula: B. Braden, "The Surveyor's Area Formula," The College Mathematics Journal, 1986. Trapezoid/trapezium naming conventions: New Zealand Mathematics Curriculum.
This calculator computes the exact area of a quadrilateral using the selected formula. For the general quadrilateral option, vertices must be entered in order (clockwise or anti-clockwise) and the shape must be non-self-intersecting. Results are in square units of whatever unit you use for the input dimensions.