How do I find the area of an irregular quadrilateral?

For an irregular quadrilateral (one that does not fit a standard shape like a rectangle or trapezoid), use the shoelace formula with the four corner coordinates listed in order (either clockwise or anti-clockwise). The formula is: Area = 0.5 x |(x1*y2 - x2*y1) + (x2*y3 - x3*y2) + (x3*y4 - x4*y3) + (x4*y1 - x1*y4)|. This works for any non-self-intersecting (simple) quadrilateral. Enter the four vertex coordinates into this calculator to get the result instantly.

What is the difference between a parallelogram and a trapezoid?

A parallelogram has two pairs of parallel sides, and opposite sides are equal in length. Its area is base x perpendicular height. A trapezoid (also called a trapezium in New Zealand and the UK) has exactly one pair of parallel sides, called the bases. Its area is 0.5 x (base1 + base2) x height. A rectangle and a rhombus are special cases of a parallelogram. A parallelogram is also a special case of a trapezoid if you use the broader definition (at least one pair of parallel sides).

Area of Quadrilateral Calculator

Calculate the area of any four-sided shape: rectangle, square, parallelogram, trapezoid (trapezium), rhombus, kite, or a general quadrilateral defined by its four corner coordinates. Select your shape type and enter the dimensions to get the area instantly with step-by-step working.

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Standard geometry formulae Area formulas verified against standard Euclidean geometry references.

1. Shape Type

Select quadrilateral type

2. Dimensions

Length (base) units

units

Width (height) units

units

Please enter valid positive numbers for all dimensions.

Area Result

Shape

Rectangle

Quadrilateral type

Area
60
Square units

Formula Used

A = l x w

Method

Step-by-Step Working

Shape Properties

Quadrilateral typeRectangle

FormulaA = length x width

Input values10 x 6

Area60 sq units

Worked Example

ShapeRectangle

Length10 units

Width6 units

FormulaA = 10 x 6

Area60 sq units

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).

Related Calculators

Maths and Stats Calculators: full collection of geometry and algebra tools.
Area Calculator: area of common shapes including circles, triangles, and rectangles.
Area of a Triangle Calculator: base-height, Heron's formula, and coordinate methods.
Area Calculator (multi-shape): calculate and add the areas of multiple shapes.
Area of a Rectangle Calculator: dedicated rectangle and square area tool.

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.

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