Quadrilaterals: In Geometry, a quadrilateral is a plane shape with four sides or edges and has four corners or vertices. Looking around, we notice many objects that have a quadrilateral shape: the floor, walls, ceiling, classroom windows, kite, chessboard, and so on. It is important to understand the concept of quadrilaterals for a thorough understanding of geometry. To help you with that, in this article, we will discuss definitions, formulas, types, shapes, and their properties in detail.

Quadrilateral Shapes Used in Daily Life

In our daily life, we see many objects that have the shape of quadrilaterals such as chessboard, playing card, kite, road sign, etc.

If we join any two points, we get a line segment, if we join three non-collinear points, we get a triangle, and if we join four points (none of the combination of three points out of these four points collinear) in order, we obtain a closed figure with four sides called a quadrilateral. They are shown in the figures below.

Quadrilateral Definition

The word quadrilateral is derived from the two Latin words: ‘quadri’ means four and ‘latus’ means sides. A quadrilateral is a two-dimensional shape having four sides, four angles, and four corners or vertices. The sum of internal angles of a quadrilateral is \({360^{\rm{o}}}.\)

Things to Know About a Quadrilateral

If \(A,\,B,\,C\) and \(D\) are co-planar points, such that,
1. No three of them are collinear.
2. The line segments \(AB,\,BC,\,CD\) and \(DA\) do not intersect except at their endpoints, then the figure made up of the four-line segments, is called quadrilateral (Abbreviation: quad).
3. The points \(A,\,B,\,C\) and \(D\) are called the vertices.
The line segment joining two non-consecutive vertices is called a diagonal.

Adjacent Sides of Quadrilateral:

The two sides of a quadrilateral that have a common end or vertex points are called its adjacent sides. In the given above figure, there are \(4\) pairs of adjacent sides. These are \(\left( {AB,\,BC} \right),\,\left( {BC,\,CD} \right),\,\left( {CD,\,DA} \right)\) and \(\left( {DA,\,AB} \right).\)

Opposite Sides of Quadrilateral:

The two sides of a quadrilateral are called its opposite sides if they do not have a common endpoint. In the given above figure, there are \(2\) pairs of opposite sides. These are \(\left( {DA,\,CB} \right)\) and \(\left( {CD,\,BA} \right).\)

Adjacent Angles of Quadrilateral:

The two angles of a quadrilateral having a common arm are called its adjacent angles. In the given above figure, there are \(4\) pairs of adjacent angles. These are \(\left( {\angle D,\,\angle A} \right),\,\left( {\angle A,\,\angle B} \right),\,\left( {\angle B,\,\angle C} \right)\) and \(\left( {\angle C,\,\angle D} \right).\)

Opposite Angles of Quadrilateral:

The two angles of a quadrilateral that are not adjacent angles are known as opposite angles. In the given above figure, there are \(2\) pairs of opposite angles. These are \(\left( {\angle A,\,\angle C} \right)\) and \(\left( {\angle B,\,\angle D} \right).\)

The Interior and Exterior of a Quadrilateral

Look at the figure above, the quadrilateral \(ABCD\) divides all the points of the plane into three parts.

The part of the plane consists of all points such as \(P,\,Q,\,R\) which lies inside the quadrilateral. This part is called the interior part of the quadrilateral. All the points like \(P,\,Q,\,R\) are called the interior points of the quadrilateral.

The part of the plane consists of all points such as \(L,\,M,\,N\) which lie outside of the quadrilateral. This part is called the exterior part of the quadrilateral. All points like \(L,\,M,\,N\) are called the exterior points of the quadrilateral.

The sides of the quadrilateral form the boundary of its interior. There are points on the boundary of the quadrilateral also.

What are the Types of Quadrilaterals?

There are many types of quadrilaterals and some special types of quadrilaterals are discussed below:

Square

A quadrilateral whose all the sides are equal and each angle measures \({90^{\rm{o}}}\) is called a square.

Rectangle

A quadrilateral in which opposite sides are of equal length and each angle is a right angle is called a rectangle.

Rhombus

A quadrilateral whose all sides are equal is called a rhombus.

Parallelogram

A quadrilateral is called a parallelogram if both pairs of its opposite sides are parallel.

Trapezium

The type of quadrilateral having exactly one pair of parallel sides is called the trapezium.

Kite

A quadrilateral is called a kite if it has two pairs of equal adjacent sides but unequal opposite sides.

How to Represent Quadrilaterals in Venn Diagram?

Quadrilaterals can be classified into squares, rectangles, rhombus, parallelograms, trapezium, and kites. Some of the properties are common in a few quadrilaterals. This can be represented with the Venn Diagram as given below:

Concave Quadrilateral vs Convex Quadrilateral

Concave Quadrilateral: If any one of the diagonals is outside the figure is called the concave quadrilateral.

Example: In the quadrilateral \(ABCD\) shown below, the diagonal \(AD\) is inside the quadrilateral, but the diagonal \(BC\) is outside the quadrilateral. Hence, \(ABCD\) is a concave quadrilateral.

Convex quadrilateral: If both diagonals are inside the figure, then it is called the convex quadrilateral.

Example: In the quadrilateral \(ABCD\) shown below, both the diagonals \(AD\) and \(BC\) lie inside the quadrilateral. Hence, \(ABCD\) is a convex quadrilateral.

Perimeter of a Quadrilateral

The total length around any two-dimensional shape is called its perimeter. To find the perimeter of any plane shape, add the length of all the sides. The quadrilateral has four sides. So, add the lengths of four sides to get the perimeter.

Suppose \(ABCD\) is a quadrilateral, perimeter of \(ABCD\) is the sum of all its sides.

Perimeter \( = AB + BC + CD + DA\)

Quadrilateral Formulas

Perimeter formulas of quadrilaterals are tabulated below:

Name of Quadrilateral	Perimeter Formula
Rhombus	\(4 \times {\rm{side}}\)
Square	\(4 \times {\rm{side}}\)
Rectangle	\({\rm{Base}} \times {\rm{Height}}\)
Parallelogram	\(2 \times \left( {{\rm{sum}}\,{\mkern 1mu} {\rm{of}}{\mkern 1mu} {\rm{the}}\,{\mkern 1mu} {\rm{adjacent}}\,{\mkern 1mu} {\rm{sides}}} \right)\)
Kite	\(2 \times \left( {a + b} \right),\) Where \(a\) and \(b\) are the lengths of all the unequal sides.

Area Formula of Quadrilaterals

The area of any shape is the space covered by it. The formula of area of some quadrilaterals are given below:

Name of Quadrilateral	Area Formula
Square	\({\rm{side}} \times {\rm{side}}\)
Rectangle	\({\rm{Length}} \times {\rm{Breadth}}\)
Parallelogram	\({\rm{Base}} \times {\rm{Height}}\)
Rhombus	\(\frac{1}{2} \times {\text{Product}}\,{\text{of}}\,{\text{the}}\,{\text{diagonals}}\)
Kite	\(\frac{1}{2} \times {\text{Product}}\,{\text{of}}\,{\text{the}}\,{\text{diagonals}}\)

What are the Properties of Quadrilaterals?

A quadrilateral is a closed two-dimensional shape formed by joining four points, among which any three points are not collinear. A quadrilateral has four sides, four angles, and four vertices or corners.

Let us understand the properties of quadrilaterals with the help of an example,

Let \({{PQRS}}\) is a quadrilateral. ( we should name the quadrilateral in order as \(PQRS,{\rm{ }}QRSP,{\rm{ }}RSPQ,{\rm{ }}SPQR\) but not \(PSQR,{\rm{ }}SRPQ,{\rm{ }}RSQP,{\rm{ }}SPRQ\)).

1. \(PQ,\,QR,\,RS\) and \(SP\) are the \(4\) sides.
2. Points \(P,\,Q,\,R\) and \(S\) are \(4\) vertices or corners.
3. \(\angle PQR,\,\angle QRS,\,\angle RSP\) and \(\angle SPQ\) are \(4\) angles.
4. \(PS\) and \(QR\) are opposite sides.
5. \(PQ\) and \(QR\) are adjacent sides.
6. \(Q\) and \(S\) are opposite angles.
7. \(Q\) and \(R\) are adjacent angles.
8. The sum of four interior angles of any quadrilateral is \({360^{\rm{o}}}.\)

Properties of a Square

1. All the sides of a square are equal.
2. Each angle measures \({90^{\rm{o}}}\) i.e., right angle.
3. The sides are parallel to each other.
4. The diagonals of a square bisect each other at \({90^{\rm{o}}}.\)

Properties of a Rectangle

1. It is a type of quadrilateral in which the opposite sides are of equal length.
2. Each angle is a right angle or \({90^{\rm{o}}}.\)
3. Opposite sides are parallel to each other.
4. The diagonals of a rectangle bisect each other.

Properties of a Parallelogram

1. A quadrilateral is called a parallelogram, if both pairs of its opposite sides are parallel and equal.
2. The diagonals of the parallelogram bisect each other.
3. The opposite angles are of equal measure.
4. The pair of adjacent angles are supplementary.

Properties of a Rhombus

1. It is a type of parallelogram having all sides equal.
2. The opposite sides are parallel to each other.
3. The opposite angles are of the same measure.
4. The total of any two adjacent angles is equal to \({180^{\rm{o}}}.\)
5. The diagonals of a rhombus bisect each other at \({90^{\rm{o}}}.\)

Properties of a Trapezium

1. The type of quadrilateral having exactly one pair of parallel sides is called the trapezium.
2. The two adjacent angles along both parallel sides of a trapezium are supplementary.
3. The diagonals of it bisect each other in the same ratio.

Properties of a Kite

1. A quadrilateral is called a kite, if it has \(2\) pairs of equal adjacent sides but unequal opposite sides.
2. The larger diagonal of kite bisects the smaller diagonal.
3. Only one pair of opposite angles are of the same measure.

Solved Examples – Quadrilaterals

Question –1: How many sides and vertices, a quadrilateral has?
Solution: A quadrilateral has \(4\) sides and \(4\) vertices.

Question –2: How many angles and diagonals, a quadrilateral has?
Solution: A quadrilateral has \(4\) angles and \(2\) diagonals.

Question- 3: What is the area of a rectangle if its length is \(9\,{\text{m}}\) and breadth is \(5\,{\text{m}}\)
Solution:
Given,
The length of the rectangle \(9\,{\text{m}}\)
The breadth of the rectangle \(5\,{\text{m}}\)
We know that the area of a rectangle \( = {\rm{Length}} \times {\rm{Breadth}}\)
Now, area of the rectangle\( = 9\,{\text{m}} \times{\text{5}}\,{\text{m}}\,{\text{=45}}\,{{\text{m}}^2}\)
Hence, the area of the rectangle is \({\text{45}}\,{{\text{m}}^2}\).

Question –4: Find the perimeter of the below figure.

Solution:
Given,
The length of the sides of the given figure are \(3\,{\text{cm,}}\,3\,{\text{cm}},\,5\,{\text{cm}}\) and \(4\,{\text{cm}}\).
We know that,
Perimeter of the figure \({{ = }}\) Sum of the length of its all the sides
Now, perimeter of \(ABCD = 3\,{\text{cm + }}\,3\,{\text{cm}} + 5\,{\text{cm}} + 4\,{\text{cm=15}}\,{\text{cm}}\)
Hence, the perimeter of the given figure is \({\text{15}}\,{\text{cm}}\).

Question -5: If the side of a square is \(6\,{\text{cm}}\) find its area.
Solution:
Given,
The length of the side of a square \(= 6\,{\text{cm}}\)
We know that the area of a square \({\rm{=side}} \times {\rm{side}}\)
Now, the area \( = 6\,{\text{cm}} \times 6\,{\text{cm}} = 36\,{{\text{cm}}^2}\)
Hence, the area of the square is \(36\,{{\text{cm}}^2}\).

Summary

In this article, we have covered some real-life examples of quadrilateral, the definition of quadrilateral, some more things about the quadrilateral. We hope going through the articles gives you a clear idea about quadrilaterals and helps you in understanding geometry.

Important Facts Related to Quadrilaterals

Here are some of the important facts related to quadrilaterals:

(i) The sum of the four angles of a quadrilateral is 360 degrees, that is, equal to four right angles or 2π radians.
(ii) The diagonals of a rectangle are equal but are not perpendicular to each other.
(iii) The diagonals of a rhombus are not equal but are perpendicular to each other.
(iv) None of the angles of the polygon are reused.
(v) The sum of the total interior angles of a pentagon is 540 degrees i.e. 6 right angles.
(vi) The number of diagonals in a pentagon is 5.
(vii) The sum of the total interior angles of a hexagon is 720 degrees i.e. 8 right angles.
(viii) The sum of the total interior angles of an octagon is 1080 degrees i.e. 12 right angles.

FAQs

Question –1: What are the six types of quadrilaterals?
Answer: There are six types of quadrilaterals. These are,
1. Square
2. Rectangle
3. Rhombus
4. Trapezium
5. Parallelogram
6. Kite

Question –2: What does quadrilateral mean?
Answer: The word quadrilateral is derived from the two Latin words ‘quadri’ means four and ‘latus’ means sides. So, a quadrilateral is a plane closed figure having four sides. 

Question –3: What are the 4 properties of a quadrilateral?
Answer: A quadrilateral is a two-Dimensional closed figure having four arms or edges or sides. The \(4\) properties of a quadrilateral are,
1. It has \(4\) sides.
2. It has \(4\) angles.
3. It has \(4\) vertices or corners.
The sum of four interior angles is \({360^{{o}}}.\)

Question –4: What shapes are quadrilaterals?
Answer: A quadrilateral is a \(4\)-sided two-dimensional shape. Some of the \(2\,D\) shapes of quadrilaterals are square, rectangle, rhombus, parallelogram, trapezium, and kite.

Question –5: How do you identify a quadrilateral?
Answer: A quadrilateral is identified by a two-dimensional closed figure having \(4\) sides, \(4\) corners, \(4\) angles and the sum of four interior angles are \({360^{{o}}}.\)

Question –6: What is the best definition of a quadrilateral?
Answer: The definition of a quadrilateral is given below along with a figure to understand clearly.

If \(A,{\rm{ }}B,{\rm{ }}C\) and \(D\) are co-planar points, such that,
1. No three of them are collinear.
2. If the line segments \(AB,{\rm{ }}BC,{\rm{ }}CD\) and \(DA\) do not intersect except at their end points, then the figure made up of the four-line segments, is called quadrilateral (Abbreviation: quad).
3. The points \(A,{\rm{ }}B,{\rm{ }}C\) and \(D\) are called the vertices.

We hope you find this article on Quadrilaterals helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.

Happy Embibing!