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Quadrilateral – Definition, Properties, Types, Formulas, 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.

Properties of Quadrilateral

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°.

Types of Quadrilateral

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

Shapes of Quadrilaterals

Quadrilaterals exhibit diverse shapes, ranging from the symmetrical squares and rectangles to the more complex and irregular parallelograms and trapezoids.

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.

Area of Trapezium

The area of a trapezium can be calculated using the formula:

Area =1/2 ​× (a+b) × (h)

Where the parallel sides (a and b) are the two sides of the trapezium that are parallel to each other, and the height (h) is the perpendicular distance between these parallel sides.

Perimeter of Trapezium

The perimeter of a trapezium is the sum of the lengths of all its four sides.

If the trapezium has side lengths a, b, c, and d, then the perimeter is:

Perimeter = a+b+c+d

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.

Area of a Kite

The area of a kite is calculated using the formula:

Area = ½ (d1 x d2)

where, d1 and d2 represent the lengths of the diagonals of the kite.

It’s important to note that this formula works regardless of whether d1 is longer or shorter than d2.

Perimeter of a Kite

The perimeter of a kite, which is the total length of all its sides added together, is calculated using the formula:

Perimeter = 2(a + b)

where, a and b represent the lengths of the two pairs of equal sides of the kite.

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.

Area of a Parallelogram

The area of a parallelogram is calculated as the product of its base and height. The formula is given as:

Area of Parallelogram ​= base × height

Where the base is any one of the parallel sides, and the height is the perpendicular distance between the base and its opposite side.

Perimeter of a Parallelogram

The perimeter of a parallelogram is the sum of the lengths of all its sides. Since opposite sides of a parallelogram are equal in length, the perimeter is calculated as:

Perimeter of Parallelogram​ = 2 × (a+b)

Where, a and b are the adjacent sides of a parallelogram.

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.

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.

Area of a Rectangle

The area of a rectangle is calculated as the product of its length and width. The formula is given as:

Area of Rectangle​ = length × width

Where the length is the longer side and the width is the shorter side of the rectangle.

Perimeter of a Rectangle

The perimeter of a rectangle is the sum of the lengths of all its sides. Since opposite sides of a rectangle are equal in length, the perimeter is calculated as:

Perimeter of Rectangle​ = 2 × (length + width)

This formula doubles the sum of the length and width because each side is counted twice.

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°.

Area of a Rhombus

The area of a rhombus can be calculated by multiplying the lengths of its two diagonals and dividing by 2. The formula is:

Area of Rhombus ​= 1/2 ​× (diagonal1 × diagonal2​)

Where, diagonal1 and diagonal2​ are the lengths of the two diagonals.

Perimeter of a Rhombus

The perimeter of a rhombus is the sum of the lengths of all its four sides. Since all sides of a rhombus are equal in length, the perimeter can be calculated as:

Perimeter of Rhombus​ = 4 × side

Where sideside is the length of any one of the sides.

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

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.

Area of a Square

The area of a square is calculated by squaring the length of one of its sides. The formula is:

Area of Square​ = side2

Where sideside is the length of one side of the square.

Perimeter of a Square

The perimeter of a square is the sum of the lengths of all its sides. Since all sides of a square are equal in length, the perimeter can be calculated as:

Perimeter of Square​ = 4 × side

Where sideside is the length of any one of the sides.

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 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

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 Shape Perimeter Formula
Parallelogram 2 (Base + Side)
Rectangle 2 (Length + Breadth)
Rhombus 4 × Side
Square  4 × Side
Kite 2 (a + b), where a and b are adjacent pairs.

Difference between Different Quadrilaterals

Square vs Rhombus

A square is a special type of rhombus where all angles are right angles and all sides are equal.

Square vs Rectangle

While both have opposite sides equal and all angles as right angles, squares have equal sides, while rectangles may have unequal sides.

Rectangle vs Parallelogram

Rectangles have all angles as right angles, unlike parallelograms which may not, but both have opposite sides equal.

Rhombus vs Kite

A rhombus has all sides equal, whereas a kite typically has two pairs of adjacent sides equal.

Rhombus vs Trapezium

Rhombuses have all sides equal, while trapeziums have one pair of parallel sides.

Rhombus vs Diamond

A diamond, when referred to geometrically, essentially represents a rhombus, sharing the same properties of equal sides and opposite angles.

Solved Questions on Quadrilateral

Here are some solved examples on quadrilaterals for your help.

Question 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).

Question 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.

Question 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°.

Related Articles

Types of Polygons

Area of a Quadrilateral

Construction of a Quadrilateral

Area of Cyclic Quadrilateral

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 : 01 Mar, 2024
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