Quadrilaterals


A Quadrilateral is a closed figure with four sides, four corners and four vertices and the sum of the interior angles is 360°.

Quadrilateral

In the above figure, ABCD is a quadrilateral with four sides AB,BC, CD and DA, and the four angles $\angle$A, $\angle$B, $\angle$C and $\angle$D formed by the four vertices A,B,C and D.

Some of the examples of quadrilaterals are classroom board, a book, table etc.

 

1. Properties of a Parallelogram:
          In a parallelogram,
Parallelogram           
  • The opposite sides are parallel to each other.
  • Opposite sides are congruent.
  • Opposite angles are congruent.
  • The diagonals of a parallelogram bisect each other.
  • Any pair of consecutive angles are supplementary (their sum is 180 degrees).

2. Properties of a Rectangle:
          In a rectangle,
Rectangle 
  • The opposite sides are parallel to each other.
  • Opposite sides are congruent.
  • Opposite angles are congruent.
  • The diagonals bisect each other.
  • All the angles are right angles.
  • The diagonals are congruent.

3. Properties of Kites:
          In a Kite,
Kite 
  • Two disjoint pairs of consecutive sides are congruent.
  • The diagonals are perpendicular to each other.
  • One diagonal is the perpendicular bisector of the other diagonal.
  • One of the diagonals bisects a pair of opposite angles.
  • One pair of opposite angles are congruent.

4. Properties of a Rhombus:
          In a Rhombus,
Rhombus             
  • The opposite sides are parallel to each other.
  • The diagonals of a parallelogram bisect each other.
  • Two consecutive sides are congruent by definition.
  • All sides are congruent.
  • All the four sides are right angles.
  • The diagonals are perpendicular bisectors of each other.
  • The diagonal divides the Rhombus into four congruent right triangles.

5. Properties of a Square:
           In a Square,
Square                 
  • The opposite sides are parallel to each other.
  • Opposite sides are congruent.
  • Opposite angles are congruent.
  • The diagonals bisect each other.
  • The diagonals are congruent.
  • All sides are congruent.
  • All the four angles are right angles.
  • The diagonals are perpendicular bisectors of each other.
  • The diagonals form four isosceles right triangles.

6. Properties of an Isosceles Trapezoid:
         In an Isosceles Trapezoid,
Isosceles Trapezoid             
  • The legs are of an Isosceles trapezoid are congruent.
  • The bases are parallel to each other.
  • The lower base and upper base angles are congruent.
  • The diagonals are congruent.
  • Any lower base angle is supplementary to any of the upper base angles.

Types of Quadrilaterals

The different types of Quadrilaterals
  • Parallelogram
  • Rectangle
  • Kite
  • Rhombus
  • Square
  • Isosceles Trapezoid
Rectangle: A rectangle is a quadrilateral in which the opposite sides are parallel and congruent, also the four angles are right angles.
Rectangle Quadrilateral
Rhombus: A Rhombus is a quadrilateral in which all the four sides are congruent, opposite sides are parallel and the opposite angles are equal (In the figure shown angles $\angle a = \angle a$ are same and angles $\angle b = \angle b$ are same).

Rhombus Quadrilateral

Square: A Square is a quadrilateral in which all the sides are congruent, all the four angles are right angles and opposite sides are parallel.

Parallelogram: A Parallelogram is a quadrilateral in which opposite sides are parallel and are equal in length, and also the opposite angles are congruent (In the figure angles $\angle x = \angle x$ are same and angles $\angle y = \angle y$ are same).

Parallelogram Quadrilateral

Rectangles, Rhombuses and Squares all different types of Parallelograms.
Kite: A Kite is a quadrilateral in which there are two pairs of sides, each pair is made by the adjacent sides that are equal in length. The angles are equal where the equal sides pair meet. Diagonals are at right angles and one of the diagonal bisects the other.
Kite
Trapezoid: A Trapezoid (Trapezium) is a quadrilateral in which one pair of opposite sides are parallel.
Trapezoid
Isosceles Trapezoid: An Isosceles Trapezoid (Trapezium) is a quadrilateral in which the sides that are not parallel are equal in length and both the angles of the equal length are equal in measure.
Isosceles Trapezoid

Area of a Quadrilateral

1.Area of a Parallelogram = base $\times$ height

                                    A = b.h
Example : Find the area of a parallelogram with base 14cm and height is 6 cm

Solution:
Given base = 14cm, Height = 6 cm

Area of a parallelogram = base $\times$ height
                                = 14 $\times$ 6
                                = 84 sq cm

2.Area of a Rectangle = length $\times$ width
          
                               A = l . w

Example : Given the length and width of a rectangle as 11m and 8 m. Find the area of the rectangle

Solution:
Given, length = 11 m,  Width = 8 m

Area of a rectangle = length $\times$ width
                           = 11 $\times$ 8
                           = 88 sq m

3.Area of a Kite = $\frac{1}{2}$ (product of the diagonals)

             If the diagonals are given as d1 and d2   

            Area of a Kite =$\frac{1}{2}$ (d1 $\times$ d2)   
Example: Find the area of a kite with diagonal lengths d1 = 26cm d2= 18cm

Solution: Length of the diagonals are d1 = 26cm, d2= 18cm

Area of a kite =$\frac{1}{2}$ (product of the diagonals)
               = $\frac{1}{2}$ ( d1 $\times$ d2)
               = $\frac{1}{2}$ ( 26 $\times$ 18)
               = 234 sq cm
4. Area of a Rhombus = $\frac{1}{2}$ (product of the diagonals)
If the diagonals are given as d1 and d2   
    Area of a Rhombus = $\frac{1}{2}$ (d1 $\times$ d2)
Example: Find the area of a Rhombus with diagonals x = 18 cm and y = 10 cm

Solution:
Given diagonals, x = 18 cm and y = 10 cm

Area of a Rhombus =$\frac{1}{2}$ (product of the diagonals)
                           = $\frac{1}{2}$ ( x $\times$ y)
                           = $\frac{1}{2}$ (18 $\times$ 10)
                           = 90 sq cm

5. Area of a Square = a²  (side $\times$ side)
Example : Given the side of a square 8 cm, find the area of the square

Solution :
  Side of the square = 8cm

Area of a square  = side $\times$ side
                        =  8 $\times$ 8
                        = 64 sq cm

6. Area of a Trapezoid (Trapezium)
= $\frac{1}{2}$ (a+b).h
(a and b are the lengths and h is the height)
Example: If a = 9 inches b = 14 inches and height, h = 6 inches. Find the area of the Trapezium

Solution:
Given, a = 9 inches, b = 14 inches, h = 6 inches

Area of a Trapezoid = $\frac{1}{2}$ (a+b).h
                           = $\frac{1}{2}$ (9 + 14). 6
                           = 69 sq inches.

Cyclic Quadrilaterals

A cyclic quadrilateral is a quadrilateral which can be inscribed in a circle.

Cyclic Quadrilateral
 
In the figure above, ABCD is a cyclic quadrilateral inscribed in a circle with centre O
A quadrilateral is said to be cyclic quadrilateral if there is a circle passing through all its vertices
Sum of the opposite angles of a cyclic quadrilateral is 180°

Area of Quadrilateral Formula

1. Area of a Parallelogram = base $\times$ height
                                      A = b.h

2. Area of a Rectangle =
length $\times$ width
                               A = l . w

3. Area of a Kite = $\frac{1}{2}$ (product of the diagonals)
    If the diagonals are given as d1 and d2   
    Area of a Kite = $\frac{1}{2}$ (d1 $\times$ d2)
          
4. Area of a Rhombus = $\frac{1}{2}$ (product of the diagonals)
  If the diagonals are given as d1 and d2   
    Area of a Rhombus = $\frac{1}{2}$ (d1 $\times$ d2)
   
5. Area of a Square = a2  (side $\times$ side)

6. Area of a Trapezoid (Trapezium) = $\frac{1}{2}$ (a+b).h