Quadrilateral

A quadrilateral is a closed shape and a type of polygon that has four sides, four vertices and four angles. It is formed by joining four non-collinear points. The sum of interior angles of quadrilaterals is always equal to 360 degrees.

The word quadrilateral is derived from the Latin words ‘Quadra’ which means four and ‘Latus’ means ‘sides’. It is not necessary that all the four sides of a quadrilateral are equal in length. Hence, we can have different types of quadrilaterals based on sides and angles. Let us more interesting facts about quadrilaterals in this article. 

What is a Quadrilateral?

A quadrilateral is a plane figure that has four sides or edges, and also has four corners or vertices. The angles are present at the four vertices or corners of the quadrilateral. If ABCD is a quadrilateral then angles at the vertices are ∠A, ∠B, ∠C and ∠D. The sides of a quadrilateral are AB, BC, CD and DA. 

If we join the opposite vertices of the quadrilateral, we get the diagonals. In the below figure AC and BD are the diagonals of quadrilateral ABCD. 

Quadrilaterals will typically be of standard shapes with four sides like rectangle, square, trapezoid, and kite or irregular and uncharacterized as shown below:

Let’s have a look at the table below to understand what quadrilateral is.

Quadrilateral	Two-dimensional plane figure enclosed by 4 line segments Or A polygon with 4 edges and 4 vertices
Number of sides	4
Number of vertices	4
Number of diagonals	2
Sum of all interior angles	360 degrees
Sum of all exterior angles	360 degrees

Types of Quadrilaterals

The types of quadrilaterals are defined based on the measure of the angles and lengths of their sides. As the word ‘Quad’ means four, all these types of a quadrilateral have four sides, and the sum of angles of these shapes is 360 degrees. The list of types of quadrilaterals are:

1. 1. Trapezium
   2. Parallelogram
   3. Squares
   4. Rectangle
   5. Rhombus
   6. Kite

Convex, Concave and Intersecting Quadrilaterals

Another way to classify the types of quadrilaterals are:

1. Convex Quadrilaterals: Both the diagonals of a quadrilateral are completely contained within a figure.
2. Concave Quadrilaterals: At least one of the diagonals lies partly or entirely outside of the figure.
3. Intersecting Quadrilaterals: Intersecting quadrilaterals are not simple quadrilaterals in which the pair of non-adjacent sides intersect. These kinds of quadrilaterals are known as self-intersecting or crossed quadrilaterals

Below are the examples of convex, concave and intersecting quadrilaterals.

Properties of Quadrilaterals

Let us understand in a better way with the help of an example:

• It has four sides: AB, BC, CD, and DA
• It has four vertices: Points A, B, C, and D
• It has four angles: ∠ABC, ∠BCD, ∠CDA, and ∠DAB
• ∠A and ∠B are adjacent angles
• ∠A and ∠C are the opposite angles
• AB and CD are the opposite sides
• AB and BC are the adjacent sides

A quadrilateral is a 4-sided plane figure. Below are some important properties of quadrilaterals :

• Every quadrilateral has 4 vertices, 4 angles, and 4 sides
• The total of its interior angles = 360 degrees

Square Properties

• All the sides of the square are of equal measure
• The sides are parallel to each other
• All the interior angles of a square are at 90 degrees (i.e., right angle)
• The diagonals of a square perpendicular bisect each other

Rectangle Properties

• The opposite sides of a rectangle are of equal length
• The opposite sides are parallel to each other
• All the interior angles of a rectangle are 90 degrees.
• The diagonals of a rectangle bisect each other.

Rhombus Properties

• All the four sides of a rhombus are of the same measure
• The opposite sides of the rhombus are parallel to each other
• The opposite angles are of the same measure
• The sum of any two adjacent angles of a rhombus is equal to 180 degrees
• The diagonals perpendicularly bisect each others 

Parallelogram Properties

• The opposite side of the parallelogram are of the same length
• The opposite sides are parallel to each other
• The diagonals of a parallelogram bisect each other
• The opposite angles are of equal measure
• The sum of two adjacent angles of a parallelogram is equal to 180 degrees

Properties of Trapezium

• Only one pair of the opposite side of a trapezium is parallel to each other
• The two adjacent sides of a trapezium are supplementary (180 degrees)
• The diagonals of a trapezium bisect each other in the same ratio

Properties of Kite

• The pair of adjacent sides of a kite are of the same length
• The largest diagonal of a kite bisect the smallest diagonal 
• Only one pair of opposite angles are of the same measure.

A glance at the Quadrilateral Properties

Sides and Angles of Quadrilaterals

Sides and angles	Square	Rectangle	Rhombus	Parallelogram	Trapezium
All sides are equal	Yes	No	Yes	No	No
Opposite sides are parallel	Yes	Yes	Yes	Yes	Yes
Opposite sides are equal	Yes	Yes	Yes	Yes	No
All the angles are of the same measure	Yes	Yes	No	No	No
Opposite angles are of equal measure	Yes	Yes	Yes	Yes	No
Diagonals bisect each other	Yes	Yes	Yes	Yes	No
Two adjacent angles are supplementary	Yes	Yes	Yes	Yes	No

Quadrilateral Formulas

There are two basic formulas for quadrilaterals, that are:

• Area 
• Perimeter

Area of Quadrilateral

The area of the quadrilateral is the total space occupied by the figure. The area formula for the different quadrilaterals are given below:

Area of a Parallelogram	Base x Height
Area of a Rectangle	Length x Width
Area of a Square	Side x Side
Area of a Rhombus	(1/2) x Diagonal 1 x Diagonal 2
Area of a Kite	1/2 x Diagonal 1 x Diagonal 2

Perimeter of Quadrilateral

Perimeter is the total distance covered by the boundary of a 2d shape. Since we know the quadrilateral has four sides, therefore, the perimeter of any quadrilateral will be equal to the sum of the length of all four sides. If ABCD is a quadrilateral then, the perimeter of ABCD is:

Perimeter = AB + BC + CD + AD

Quadrilateral Name	Perimeter
Square	4 x Side
Rectangle	2(Length + Breadth)
Parallelogram	2(Base + Side)
Rhombus	4 x Side
Kite	2 (a + b), a and b are adjacent pairs

Important Facts of Quadrilateral

• A quadrilateral is a trapezoid or a trapezium if 2 of its sides are parallel to each other.
• A quadrilateral is a parallelogram if 2 pairs of sides parallel to each other.
• Squares and Rectangles are special types of parallelograms. Below are some special properties.– All internal angles are of “right angle” (90 degrees).– Each figure contains 4 right angles.– Sides of a square are of the same length (all sides are congruent) – Opposite sides of a rectangle are the same.– Opposite sides of a rectangle and square are parallel.
• A quadrilateral is a rhombus, if
  • All the sides are of equal length-specified 2 pairs of sides are parallel to each other.
• A kite is a special sort of quadrilateral, in which 2 pairs of adjacent sides are equal to each other.

Quadrilaterals Solved Examples

Example 1: What is the base of a rhombus, if its area is 40 square units and the height is 8 units?
Solution:
Given,
Area = 40 square units
Height = 8 units
Area of rhombus = Base × Height
40 = Base × 8
Base = 40/8 = 5 units

Example 2: If 15 metres and 6 metres are diagonal lengths of a  kite, then what is its area?

Solution: Given, diagonal 1 = 15 metre and diagonal 2 = 6 metre. So, the area is simply calculated as, (1/2)(15×6) = 45 m2.

Example 3: Find the perimeter of the quadrilateral with sides 5 cm, 7 cm, 9 cm and 11 cm.

Solution: Given, sides of a quadrilateral are 5 cm, 7 cm, 9 cm and 11 cm.

Therefore, perimeter of quadrilateral is:

P =  5 cm + 7 cm + 9 cm + 11 cm = 32 cm

Example 4: The perimeter of the quadrilateral is 50 cm and the lengths of the three sides are 9 cm, 13 cm and 17 cm. Find the missing side of the quadrilateral.

Solution: Let the unknown side of the quadrilateral = x

Given, Perimeter of the quadrilateral = 50 cm

The lengths of other three sides are 9 cm, 13 cm and 17 cm

As we know,

Perimeter = sum of all the four sides.

50 = 9 cm + 13 cm + 17 cm + x

50 = 39 + x

x = 50 – 39

x = 11

Therefore, the fourth side of quadrilateral = 11 cm 

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