Four sides of an irregular quadrilateral can be arranged in convex, concave or crossed shape.

(We assume that the vertices are connected by the sequence from A to B then to C and to D and finally back to A) Because any arbitrary 4 sides can form a convex, concave or crossed quadrilateral it is mandatory to define the exact form.

In order to draw a quadrilateral closed shape the following inequalities must be fulfilled:

a + b + c > d

b + c + d > a

c + d + a > b

d + a + b > c

Any quadrilateral shape can be divided into 2 triangles.

The area of a convex quadrilateral can be expressed in one of the following formulas:

It can be seen from Fig. 3 that folding triangle BCD along q axis forms a concave quadrilateral. The question now is how can we estimate if folding the triangle will form a concave or crossed shape. From fig. 2 we can see that if β₁> β₂ and δ₁> δ₂ are both true then the new shape will be concave else if one of the criteria is false the new shape is a crossed quadrilateral. If both criteria are false then it is a concave shape but triangle ABD is folded into triangle BCD instead.

If the vertices coordinate of a convex quadrilateral is given, then the area can be calculated easily by cross product of the vectors, formed from the two opposite vertices.

And the two opposite vectors are:

And the area is:

After solving the cross product we get a more simple form of the area, It must be remembered that each determinant should be taken as positive value even if the result is negative. (See example 1)