Quadrilateral is a polygon of four sides and four vertices. It is also called tetragon and quadrangle. For triangles, 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

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

 

where
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

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

 

Common Quadrilaterals

Square

 
square.gif

 

Area, $A = a^2$

Perimeter, $P = 4a$

Diagonal, $d = a\sqrt{2}$

 

Rectangle

 
rectangle.gif

 

Area, $A = ab$

Perimeter, $P = 2(a + b)$

Diagonal, $d = \sqrt{a^2 + b^2}$

 

Rhombus

 
rhombus.gif

 

Area, $A = a^2 \sin \theta = ah$

Perimeter, $P = 4a$

Shorter diagonal, $d_1 = a\sqrt{2 - 2 \cos \theta}$

longer diagonal, $d_2 = a\sqrt{2 + 2 \cos \theta}$

Note: The diagonals of square and rhombus are perpendicular to each other.

 

Parallelogram

 
parallelogram.gif

 

Area, $A = ab \sin \theta = ah$

Perimeter, $P = 2(a + b)$

Shorter diagonal, $d_1 = \sqrt{a^2 + b^2 - 2ab \cos \theta}$

Longer diagonal, $d_2 = \sqrt{a^2 + b^2 + 2ab \cos \theta}$

 

Trapezoid

 
trapezoid.gif

 

Area, $A = \frac{1}{2}(a + b)h$

 

The Cyclic Quadrilateral

A quadrilateral is said to be cyclic if its vertices all lie on a circle. In cyclic quadrilateral, the sum of two opposite angles is 180° (or π radian); in other words, the two opposite angles are supplementary.

$A + C = 180^\circ$

$B + D = 180^\circ$

 

Cyclic quadrilateral

 

The area of cyclic quadrilateral is given by

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

See the derivation of area of cyclic quadrilateral for profound details.

 

Ptolemy's Theorem for Cyclic Quadrilateral
For any cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of non-adjacent sides. In other words

$d_1 d_2 = ac + bd$

See the proof of Ptolemy's theorem for cyclic quadrilateral.

 

Quadrilateral Circumscribing a Circle

Quadrilateral circumscribing a circle (also called tangential quadrilateral) is a quadrangle whose sides are tangent to a circle inside it.
 

Tangential Quadrilateral

 

Area,

$A = rs$

Where r = radius of inscribed circle and s = semi-perimeter = (a + b + c + d)/2
 

Derivation for area
Let O and r be the center and radius of the inscribed circle, respectively.
 

tangential-quadrilateral-area.gif

 

$A_{AOB} = \frac{1}{2}ar$

$A_{BOC} = \frac{1}{2}br$

$A_{COD} = \frac{1}{2}cr$

$A_{AOD} = \frac{1}{2}dr$
 

Total area
$A = A_{AOB} + A_{BOC} + A_{COD} + A_{AOD}$

$A = \frac{1}{2}ar + \frac{1}{2}br + \frac{1}{2}cr + \frac{1}{2}dr$

$A = \frac{1}{2}(a + b + c + d)r$

$A = sr$       (okay!)
 

Some known properties

  1. Opposite sides subtend supplementary angles at the center of inscribed circle. From the figure above, ∠AOB + ∠COD = 180° and ∠AOD + ∠BOC = 180°.
  2. The area can be divided into four kites. See figure below.
     

    tangential-quadrilateral-kites.gif
     

  3. If the opposite angles are equal (A = C and B = D), it is a rhombus.

 

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