The Quadrilateral

Quadrilateral is a polygon of four sides and four vertices. It is also called tetragon and quadrangle. In the triangle, the sum of the interior angles is 180°; for quadrilaterals the sum of the interior angles is always equal to 360°

$A + B + C + D = 360^\circ$


Classifications of Quadrilaterals
There are two broad classifications of quadrilaterals; simple and complex. The sides of simple quadrilaterals do not cross each other while two sides of complex quadrilaterals cross each other.

Simple quadrilaterals are further classified into two: convex and concave. Convex if none of the sides pass through the quadrilateral when prolonged while concave if the prolongation of any one side will pass inside the quadrilateral.

convex quadrilateral, concave quadrilateral, and complex quadrilateral


The following formulas are applicable only to convex quadrilaterals.

General Quadrilateral

General Quadrilateral


Any convex quadrilateral can use the following formulas:

Perimeter, P (applicable to all quadrilaterals, simple and complex)

$P = a + b + c + d$


Area, A

$A = \sqrt{(s - a)(s - b)(s - c)(s - d) - abcd \cos^2 \varphi \,}$


s = semi perimeter = ½P
φ = ½ (A + C) or φ = ½ (B + D)

The area can also be expressed in terms of diagonals d1 and d2

$A = \frac{1}{2}d_1 d_2 \sin \theta$