Area of a Quadrilateral Calculator

How to Find the Area of Any Quadrilateral

A quadrilateral is a 2D geometric polygon with four edges (sides) and four vertices (corners). Unlike triangles, whose shape and area are uniquely fixed by their three side lengths via Heron's formula, four side lengths alone do not determine a unique quadrilateral. You need additional information, such as two opposite angles, diagonal lengths and their intersection angle, or the $(x, y)$ Cartesian coordinates of all four vertices.

This comprehensive calculator supports the four standard mathematical methods to compute quadrilateral area and perimeter. For specialized quadrilaterals, visit our Area of a Trapezoid Calculator, Parallelogram Calculator, and Rectangle Calculator.

Method 1: Bretschneider's Formula (General Convex Quadrilateral)

Bretschneider's formula calculates the area of any convex quadrilateral given the four side lengths $a, b, c, d$ and any pair of opposite interior angles $\alpha$ and $\gamma$:

$$K = \sqrt{(s-a)(s-b)(s-c)(s-d) - abcd \cos^2\left(\frac{\alpha + \gamma}{2}\right)}$$

where $s$ is the semi-perimeter:

$$s = \frac{a + b + c + d}{2}$$

Method 2: Brahmagupta's Formula (Cyclic Quadrilateral)

A cyclic quadrilateral is one whose four vertices all lie on a single circumference (circumcircle). In any cyclic quadrilateral, opposite angles are supplementary ($\alpha + \gamma = 180^\circ$). Because $\cos(90^\circ) = 0$, Bretschneider's formula simplifies to Brahmagupta's formula:

$$K = \sqrt{(s-a)(s-b)(s-c)(s-d)}$$

Method 3: Diagonals and Included Angle

If you know the lengths of the two interior diagonals $d_1$ and $d_2$ and the angle $\theta$ between them:

$$K = \frac{1}{2} d_1 d_2 \sin(\theta)$$

When the diagonals intersect at right angles ($\theta = 90^\circ$, such as in kites, rhombuses, and orthodiagonal quadrilaterals), $\sin(90^\circ) = 1$ and the area simplifies to $K = \frac{1}{2} d_1 d_2$.

Method 4: Vertex Coordinates (Shoelace Formula)

For a quadrilateral plotted on a 2D coordinate grid with ordered vertices $A(x_1, y_1), B(x_2, y_2), C(x_3, y_3), D(x_4, y_4)$, the area is calculated using Gauss's area formula (shoelace formula):

$$K = \frac{1}{2} |(x_1 y_2 + x_2 y_3 + x_3 y_4 + x_4 y_1) - (y_1 x_2 + y_2 x_3 + y_3 x_4 + y_4 x_1)|$$

Comparison of Quadrilateral Formulas

Method	Required Inputs	Formula	Best Suited For
Bretschneider	4 sides ($a, b, c, d$) + 2 opposite angles ($\alpha, \gamma$)	$\sqrt{\prod(s-x) - abcd\cos^2(\varphi)}$	General land surveying & irregular plots
Brahmagupta	4 sides ($a, b, c, d$)	$\sqrt{(s-a)(s-b)(s-c)(s-d)}$	Cyclic (inscribed) quadrilaterals
Diagonals	Diagonals $d_1, d_2$ + angle $\theta$	$\frac{1}{2} d_1 d_2 \sin(\theta)$	Kites, rhombuses & structural bracing
Shoelace	4 coordinates $(x_i, y_i)$	$\frac{1}{2}\|\Sigma_1 - \Sigma_2\|$	GIS mapping, CAD & graphic design

Frequently Asked Questions

Can you find the area of a quadrilateral with only 4 side lengths?

Only if the quadrilateral is cyclic (all four vertices touch a circle) or an equidiagonal shape. A generic 4-sided frame with hinges can flex into infinitely many shapes with different interior angles and different areas without changing its side lengths.

What is the maximum possible area for 4 given side lengths?

According to Brahmagupta's inequality, for any four given side lengths, the maximum possible area is achieved when the quadrilateral is cyclic ($\alpha + \gamma = 180^\circ$).

What makes a quadrilateral impossible to construct?

By the polygon inequality theorem, the longest side of a quadrilateral must be strictly shorter than the sum of the other three sides ($a < b + c + d$). If one side equals or exceeds the sum of the remaining three, the sides cannot close into a polygon.

How does the shoelace formula handle vertex order?

You must enter the four vertices consecutively in either clockwise or counter-clockwise order around the perimeter (e.g., A to B to C to D). Listing vertices out of order (such as crisscrossing diagonals) calculates an incorrect self-intersecting polygon area.

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