Today we are going to learn about quadrilaterals and their properties.The term quadrilateral is a bit difficult and you might not have heard about it or familiar with it, but you must have heard about squares and rectangles. We have been studying these since grade 2. Square and rectangle are quadrilaterals, and this is what we are going to discuss today. We are going to learn all the different types of quadrilaterals, their properties and details like area, and perimeter. We are also going to solve some example problems so when you are given some question, you will be able to solve it easily.

So, let’s get started!

**What is a quadrilateral **

The word quadrilateral means a polygon having four sides (quad means four and lateral means side). So, a quadrilateral is a 2 dimensional, closed (exactly each side meet at the vertex) plane figure having four sides and hence, four vertices (or corners).

### Properties of a quadrilateral

- The sum of the internal angles is 360
^{o}or 2π radians. - If A, B, C, and D are the vertices of a quadrilateral. Then,

∠A + ∠B +∠C + ∠D = 360^{o}

**Also know: Properties of triangle**

**Types of quadrilaterals**

There are different types of quadrilaterals. The most common ones are:

Let us talk about each type of quadrilateral in detail along with their properties.

**Kite**

A kite shape is a quadrilateral, which has two pair of sides where each pair is made up of two adjacent sides of equal length.

In above diagram AB and AD forms one pair whereas BC and CD forms the second pair.

- AB = AD
- BC = DC
- ∠ABC = ∠ADC

**Perimeter of a kite**

The perimeter is nothing but the length of the boundary of a closed figure which can be found by adding the length of sides of that figure.

So, perimeter of kite is sum of the length of all sides. i.e.

If AB, BC, CD and AD are the sides of kite. Moreover, AB = AD and BC = DC. Then,

Perimeter, p = AB + BC + CD + DA

p = 2*AB + 2*CD

p = 2 ( AB + CD )

In simple words perimeter of a kite can be given by twice of the sum of the length of any two opposite sides.

**Area of the kite**

The area of the kite can be found as follows:

If d_{1} and d_{2} are the diagonals of the kite. Then, the formula for the area of a kite =

**Area = (d _{1} × d_{2}) / 2**

So, area of a kite can be calculated as half of the multiplication of it’s diagonal.

#### Properties of a kite

- The diagonals of the kite intersect at right angles.
- Perimeter is twice of the sum of the length of any two opposite sides.
- Area is half of the multiplication of it’s diagonal.

**Parallelogram**

It is a type of quadrilateral in which the opposite sides are parallel and opposite angles are equal. Thus, it has two pairs that are parallel and are of equal lengths. The diagonals in a parallelogram bisect each other i.e. intersect each other at mid-points, not necessarily at the right angle.

If A, B, C, and D are the vertices of the parallelogram with sides AB and CD parallel to each other and AD and BC are parallel too. Then,

- AB = CD & AD = BC
- The angles, ∠A = ∠C and ∠B = ∠D.

Moreover, let P be the point where diagonals bisect. Then,

- AP = PC & BP = PD

**Perimeter of a parallelogram **

Let a be the length of the side CD or the base of the parallelogram and b be the side length of BC. Then, the formula of perimeter of a parallelogram

**Perimeter = p = 2(a + b)**

So, the perimeter of a parallelogram is the twice of the sum of the length of its opposite sides.

**Area of a parallelogram**

Area of the parallelogram is given as:

**Area = Δ = a × h**

where a is the base (length of the side CD) and h is the corresponding height or altitude.

#### Properties of parallelogram

- Opposite sides of a parallelogram are always equal and parallel to each other
- Opposite angles are equal
- Diagonal of a parallelogram bisect each other but not necessarily at a right angle
- Perimeter is equal to the two times the sum of the length of its opposite sides
- The diagonals of a parallelogram always bisect each other
- It is different from the rectangle in the sense that it does not have the right angle at each vertex, whereas a rectangle has always the right angle at the vertex.

**Rhombus**

A rhombus is a quadrilateral with all four sides congruent i.e. all the sides of equal lengths. The opposite sides are parallel to each other.

In the figure below A, B, C, and D are the vertices of the rhombus. Sides AB and CD are parallel to each other and sides AD and BC are also parallel. Then,

- AB = BC = CD = DA
- ∠A = ∠C & ∠B = ∠D.

**Perimeter of rhombus **

The formula for perimeter of rhombus is:

**p = 4a, **where a is the length of each side

#### Area of rhombus

Area of a rhombus is given by the following formula

**Area = a × h**

Where a is the base (length of the side CD) and h is the corresponding height or altitude.

**Or**

**Δ = (d1 × d2) / 2**

Where d1 and d2 are the diagonals of the rhombus.

You can observe here that both the formulas for the area are the same as that of kite and parallelogram because a rhombus is a kite and also a parallelogram.

#### Properties of rhombus

- The diagonals of the rhombus bisect each other at 90
^{o}. - All sides are equal
- Opposite angles are equal
- It is different from the square in the sense that it does not have the right angle at each vertex, whereas a square has always the right angle at each vertex.
- If all sides of a parallelogram become equal, it will be a rhombus

**Rectangle**

It is a quadrilateral with opposite sides of equal length and every angle of 90^{o}. The opposite sides are parallel to each and adjacent sides are perpendicular. It is also a parallelogram and hence it possesses all the properties of a parallelogram.

- AB = CD & AC = BD
- ∠A = ∠B = ∠C = ∠D = 90
^{0}

**Perimeter and Area of rectangle**

**Rectangle perimeter formula **

**P = 2 * ( a + b )**

Where a is the length of the sides AB or CD and b is the length of the sides AC or BD.

**Rectangle area formula**

**Area = a × b **

Where a is the base and b is the corresponding height, perpendicular to the base.

#### Properties of a rectangle

- Opposite sides are equal and parallel
- The angle at each vertex is equal to 90
^{0} - Product of any two adjacent sides is equal to its area
- Two diagonals bisect each other at different angles.
- Its both diagonals are always equal

**Square**

A quadrilateral with all the sides equal and every angle of 90^{o} is a square. The opposite sides in a square are parallel to each other and adjacent sides are perpendicular just like in rectangle. The diagonals intersect each other at midpoint perpendicularly.

- AB = BC = CD = DA
- ∠A = ∠B = ∠C = ∠D = 90
^{0}

**Perimeter and Area of a square**

Let a be the length of each side. Then,

**Perimeter = 4a****Area = a × h = a*a = a**^{2}

Here h is equal to a because the side with length a is perpendicular to the base. So,Area = a^{2}

#### Properties of a square

- All sides are equal
- Angle at each vertex is equal to 90
^{0} - Its diagonals are equal
- Diagonal bisect each other at right angle
- Area of a square is given by the square of it’s any side
- A square is also a rhombus, a rectangle and a parallelogram.

**Trapezium or Trapezoid**

This might be a little bit confusing because the definitions of the terms vary according to definitions of the UK or the US.

In the US, a trapezoid is a quadrilateral in which at least one pair of parallel sides exist, and a trapezium is a quadrilateral that has no parallel sides. The definitions of the UK are the opposite. So, a trapezoid has no parallel sides and trapezium has at least one pair of parallel sides. In India the UK definition is followed.

**Perimeter and Area of a trapezium**

- Perimeter = p = AB + BC + CD + AD
- Area = h*(AB + CD) / 2

Where AB and CD are parallel to each other and h (height) is the perpendicular distance between AB and CD.

So, the area of trapezium is calculated by the sum of it’s two parallel sides multiplied by the half of the perpendicular distance between them.

#### Properties of a trapezium

- At-least one pair of opposites sides are parallel

**Isosceles Trapezium**

A trapezium is an isosceles trapezium that has non-parallel sides of equal size and angle.

**Concave and Convex Quadrilaterals**

As you have seen many types of quadrilaterals but one of the classifications of quadrilaterals can be a concave and convex quadrilateral. Concave quadrilaterals are those quadrilaterals in which at least one of the interior angles is greater than 180^{o} or **at least one of the diagonals goes outside the figure**. On the other hand convex quadrilaterals are those quadrilaterals in which each of the interior angles is less than 180^{o }or **both the diagonals are completely contained inside the quadrilateral**.

**Self-Intersecting Quadrilateral**

Self-intersecting quadrilaterals or simply intersecting or complex quadrilaterals are those quadrilaterals in which two non-adjacent sides intersect.

**Now you have learnt all types of quadrilaterals and their properties. Its time to solve some questions.**

**Example 1**

Adam wants to tile the floor of his room. If the cost of the tiles per square meter is Rs. 20. What will be the total cost, if the floor is rectangular having the base of 40 meters and a side length of 20 meters?

**Solution**

Finding the Area of the floor using formula

formula for area of a rectangle = a × b

base = a = 40 m

side length = b = 20 m

Area = 40 × 20 = 800 m^{2}

If the cost per square meter is Rs. 20 then, **the total price will be 800 × 20 = Rs. 16, 000**

**Example 2**

If the area of the parallelogram of height 20 cm is 80 cm^{2 }and the length of a single side AC is 15 cm. Find the perimeter.

**Solution**

Area = 80 cm^{2 }

Side length = b = 15 cm

Height = h = 20 cm

Using the area formula

Area of a parallelogram = a × h

80 = a × 20

a = 4 cm

The base is of length 4 cm.

To find perimeter, using formula

Perimeter of a rectangle = 2(a+b)

p = 2*( 4+15 )

p = 2 * 19

p =38 cm

**So, the perimeter of parallelogram is 38 cm.**