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Quadrilateral | Definition, Properties, Types, Examples

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Quadrilateral is a type of polygon that has four sides, four vertices, and four angles, along with a pair of diagonals. The sum of its interior angles is 360°. Let’s learn what is a quadrilateral, its types, properties, and examples in detail.

Quadrilateral Diagram
Illustration of a Quadrilateral


Quadrilateral Definition

A quadrilateral is defined as a polygon having 4 sides, 4 angles, and 4 vertices. The quadrilateral is a type of polygon in which the sides are defined in a proper pattern.

For example, in the diagram below, the quadrilateral can be defined as ABCD, ADCB, BCDA, CDAB, etc. It cannot be defined as ACBD or BDAC. Here, the quadrilateral’s sides are AB, BC, CD, and DA, and the diagonals are AC and BD.

Quadrilateral Properties

The properties of a quadrilateral are:

  • A quadrilateral has 4 sides.
  • It has 4 vertices.
  • It has 4 angles.
  • It has 2 diagonals.
  • The sum of its internal angles is 360°.

Quadrilateral Types

Based on their properties, quadrilaterals are divided into two major types:

  1. Convex quadrilaterals and
  2. Concave quadrilaterals.

These concave and convex quadrilaterals can be further classified into their subdivisions.

Types of Quadrilateral
Shape of Different Quadrilaterals

Concave Quadrilateral

Quadrilaterals that have one interior angle greater than 180° and one diagonal lies outside the quadrilateral are called concave quadrilaterals.

One of the examples of a concave quadrilateral is a Dart. It is a quadrilateral with bilateral symmetry like a kite, but with a reflex interior angle.

Dart- A Concave Quadrilateral
Diagram of a Dart

Here, in the image given below, one of the interior angles of the quadrilateral is 210°, which is greater than 180°. Therefore,  the quadrilateral is a concave quadrilateral.

Convex Quadrilateral

Quadrilaterals that have all four interior angles less than 180° are called concave quadrilaterals.

There are various types of Convex Quadrilaterals, which are :

  1. Trapezium
  2. Kite
  3. Parallelogram
  4. Rectangle
  5. Rhombus
  6. Square
Convex Quadrilateral Diagram
Convex Quadrilateral

Trapezium

A trapezium is a quadrilateral that has one pair of opposite sides parallel. In a regular trapezium, non-parallel sides are equal, and its base angles are equal.

The area of trapezium is 1/2 × Sum of parallel sides × Distance between them.

Trapezium
Trapezium Illustration

Properties of Trapezium

Here are two important properties of a trapezium:

  • The sides of the trapezium that are parallel to each other are known as the bases of trapezium. In the above image, AB and CD are the base of the trapezium.
  • The sides of the trapezium that are non-parallel are called the legs. In the above image, AD and BC are the legs.

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Kite

Kite has two pairs of equal adjacent sides and one pair of opposite angles equal. Diagonals of kites intersect perpendicularly.

The longest diagonal of the kite bisects the smaller one.

Kite Diagram
Kite Illustration

Properties of Kite

Let’s discuss some of the properties of a kite.

  • A kite has two pairs of equal adjacent sides. For example, AC = BC and AD = BD.
  • The interior opposite angles that are obtuse are equal; here, ∠A = ∠B.
  • The diagonals of a kite are perpendicular to each other; here, AB is perpendicular to CD.
  • The longer diagonal of the kite bisects the shorter diagonal. Here, CD bisects AB.

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Parallelogram

Parallelogram is a quadrilateral whose opposite sides are equal and parallel. Opposite angles of a Parallelogram are equal, and its diagonals bisect each other.

Parallelogram Diagram
Parallelogram Illustration

Properties of Parallelogram

The properties of a Parallelogram are:

  • The opposite sides of a parallelogram are parallel and equal. In the above example, AB and CD are parallel and equal, and AC and BD are parallel and equal.
  • The opposite angles in a parallelogram are equal. ∠A = ∠D and ∠B = ∠C.
  • The diagonals of a parallelogram bisect each other.

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Rectangle

Rectangle is a quadrilateral whose opposite sides are equal and parallel and all the interior angles equal to 90°.

Diagonals of a rectangle bisect each other.

The area of rectangle is length × breadth.

Rectangle Diagram
Illustration of a Rectangle

Note that all the rectangles are parallelograms, but the reverse of this is not true.

Rectangle Properties

These are some of the important properties of rectangle:

  • The opposite sides of a rectangle are parallel and equal. In the above example, AB and CD are parallel and equal, and AC and BD are parallel and equal.
  • All 4 angles of a rectangle are equal and are equal to 90°. ∠A = ∠B = ∠C = ∠D = 90°.
  • The diagonals of a rectangle bisect each other and the diagonals of a rectangle are equal, here, AD = BC.

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Rhombus

Rhombus is a quadrilateral that has all sides equal and opposite sides parallel. Opposite angles of a rhombus are equal, and diagonals of the Rhombus bisect each other perpendicularly. .

Rhombus Diagram
Rhombus Diagram

Note all rhombus are parallelograms, but the reverse of this is not true.

Properties of Rhombus

Here are some of the key properties of a Rhombus:

  • All 4 sides of a rhombus are equal. AB = BC = CD = AD.
  • The opposite sides of a rhombus are parallel and equal. In the image above, AB is parallel to CD and AD is parallel to BC.
  • The diagonals of a rhombus Bisect each other at 90°.

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Square

A quadrilateral that has all sides equal and opposite sides parallel and all interior angles equal to 90° is called a Diagonals of squares bisect each other perpendicularly. Note that all squares are rhombus but not vice-versa. 

Diagram of Square

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Properties of Square

The properties of a square are:

  • All four sides of a square are equal to each other.
  • The interior angles of a square are 90°.
  • The diagonal of a square bisects each other at 90°.
  • The opposite sides are parallel, and the adjacent sides are perpendicular in a square.

Quadrilateral Sides and Angles

The following table illustrates how the sides and angles of quadrilaterals make them different from one another:

Characteristics of Quadrilaterals

Sides and anglesSquareRectangle Rhombus ParallelogramTrapezium
All sides are equalYesNoYesNoNo
Opposite sides are parallelYesYesYesYesYes
Opposite sides are equalYesYesYesYesNo
All the angles are of the same measureYesYesNoNoNo
Opposite angles are of equal measureYesYesYesYesNo
Diagonals bisect each otherYesYesYesYesNo
Two adjacent angles are supplementaryYesYesYesYesNo

Quadrilateral Formulas

The two basic formulas for quadrilaterals, are:

  • Area of Quadrilateral
  • Perimeter of Quadrilateral

Let’s discuss them in detail.

Area of Quadrilateral

The area of a quadrilateral is defined as the amount of space covered by a quadrilateral in a two-dimensional space. The different types of quadrilaterals have different formulae for their respective area.

The chart given below shows the formula for the area of each type of quadrilateral:

Calculating Quadrilateral Area

Perimeter of Quadrilateral

The perimeter of a quadrilateral is the sum of all its sides. For different quadrilateral shapes, the perimeter formula is different as they portray different characteristics.

Below are the formulas for different quadrilateral shapes.

Perimeter Formula of Quadrilaterals

Quadrilateral ShapePerimeter Formula
Parallelogram2 (Base + Side)
Rectangle2 (Length + Breadth)
Rhombus4 × Side
Square 4 × Side
Kite2 (a + b), where a and b are adjacent pairs.

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Solved Examples on Quadrilateral

Here are some solved examples on quadrilaterals for your help.

Example 1: The perimeter of quadrilateral ABCD is 46 units. AB = x + 7, BC = 2x + 3, CD = 3x – 8, and DA = 4x – 6. Find the length of the shortest side of the quadrilateral. 

Solution

Perimeter = Sum of all sides

= 46 = 10x – 4 or [x = 5]

That gives, AB = 12 units, BC = 13 units, CD = 7 units, DC = 14 units

Hence, length of shortest side is 7 units (i.e. CD).

Example 2: Given a trapezoid ABCD (AB || DC) with median EF. AB = 3x – 5, CD = 2x -1 and EF = 2x + 1. Find the value of EF.

Solution

We know that the Median of the trapezoid is half the sum of its bases.

= EF = (AB + CD) / 2

= 4x + 2 = 5x – 6  or [x = 8]

Therefore EF = 2x + 1 = 2(8) + 1 => EF = 17 units.

Example 3: In a Parallelogram, adjacent angles are in the ratio of 1:2. Find the measures of all angles of this Parallelogram.

Solution: 

Let the adjacent angle be x and 2x.

We know that in of a Parallelogram adjacent angles are supplementary.

Quadrilateral Solved Example

= x + 2x = 180° or [x = 60°]

Also, opposite angles are equal in a Parallelogram.

Therefore measures of each angles are 60°, 120°, 60°, 120°.

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FAQs on Quadrilateral

What is Quadrilateral?

A quadrilateral can be defined as a type of polygon that has four sides, four vertices, and four angles along with a pair of diagonals. The Sum of interior angles of a quadrilateral is 360°.

How Many Types of Quadrilateral are there?

Here are the main types of quadrilaterals:

  1. Square: All sides are equal, and all angles are right angles (90 degrees).
  2. Rectangle: Opposite sides are equal and parallel, and all angles are right angles.
  3. Rhombus: All sides are equal, but angles are not necessarily right angles. Opposite angles are equal, and opposite sides are parallel.
  4. Parallelogram: Opposite sides are equal and parallel. Opposite angles are equal, but not all angles have to be right angles.
  5. Trapezium : At least one pair of opposite sides is parallel. A trapezium has at least one pair of parallel sides.
  6. Kite: Two pairs of adjacent sides are of equal length. One diagonal bisects the other at right angles.

What is the Sum of Interior Angles of Quadrilateral?

The sum of the interior angles of a quadrilateral is 360°. The same case applies to all quadrilateral shapes.

How To Find the Area of Quadrilateral?

The area of a quadrilateral is defined as the space covered by the quadrilateral shape in 2-dimensional space. The different types of quadrilateral have different formula for area according to their properties. For example, the area of square is a2 where, a is the side length, the area of rectangle is l × b, where l and b are the length and breadth respectively, etc.

What is Cyclic Quadrilateral?

A cyclic quadrilateral is defined as that quadrilateral in which all the four vertices of the quadrilateral lie on the circumference of a circle. 

What are Properties of Quadrilateral?

Here are the key properties of a quadrilateral:

  • Four Sides: By definition, all quadrilaterals have four straight sides.
  • Four Vertices: The points where the sides meet are called vertices, and quadrilaterals have four of these.
  • Four Angles: There are four interior angles in a quadrilateral.
  • Sum of Interior Angles: The sum of the interior angles in any quadrilateral is 360 degrees.
  • Two Diagonals: Quadrilaterals have two diagonals, which are segments connecting opposite vertices.

Explain the Golden Quadrilateral.

Golden quadrilateral is the famous term given to the national highway that connects the major states and the industrial, agriculateral, and cultural centres of India. On the map, the shape of the highway is the shape of a quadrilateral.

Which Quadrilateral is not a Parallelogram?

A trapezium is a quadrilateral that is not a parallelogram.



Last Updated : 14 Feb, 2024
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