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This pdf includes the following topics:-

Trapezium Definition

Trapezium Basic Concept

Properties of a Trapezium

Area of a Trapezium

Trapezium Definition

Trapezium Basic Concept

Properties of a Trapezium

Area of a Trapezium

1.
Week 1 & 2

Trapezum

Area of trapezium is the region covered by a trapezium in a two-

dimensional plane. It is the space enclosed in 2D geometry. A

trapezium is a 2D shape which falls under category of

quadrilaterals. Similar to other geometrical shapes, it also has its

own properties and formulas based on area and perimeter. Let us

learn in detail.

Trapezium Definition

A trapezium is a quadrilateral, which is defined as a shape with

four sides and one set of parallel sides. Apart from trapezium,

there are four more types of quadrilaterals. They are:

1 Parallelogram

2 Rectangle

3 Square

4 Rhombus

All these quadrilaterals have one common property, which is, the

sum of all the angles is 3600.

Trapezium Basic Concept

1 The pair of parallel sides are called the bases while the non-

parallel sides are called the legs of the trapezoid

2 The line segment connecting the midpoints of the non-parallel

sides of a trapezoid is called the mid-segment

3 Check above the different types of trapezium figures, where

arrow represents the parallel side of it. In all the three figures

you can see, the two sides are parallel to each other,

whereas the other two sides are non-parallel

Trapezum

Area of trapezium is the region covered by a trapezium in a two-

dimensional plane. It is the space enclosed in 2D geometry. A

trapezium is a 2D shape which falls under category of

quadrilaterals. Similar to other geometrical shapes, it also has its

own properties and formulas based on area and perimeter. Let us

learn in detail.

Trapezium Definition

A trapezium is a quadrilateral, which is defined as a shape with

four sides and one set of parallel sides. Apart from trapezium,

there are four more types of quadrilaterals. They are:

1 Parallelogram

2 Rectangle

3 Square

4 Rhombus

All these quadrilaterals have one common property, which is, the

sum of all the angles is 3600.

Trapezium Basic Concept

1 The pair of parallel sides are called the bases while the non-

parallel sides are called the legs of the trapezoid

2 The line segment connecting the midpoints of the non-parallel

sides of a trapezoid is called the mid-segment

3 Check above the different types of trapezium figures, where

arrow represents the parallel side of it. In all the three figures

you can see, the two sides are parallel to each other,

whereas the other two sides are non-parallel

2.
4 If we draw a line segment, between the two non-parallel sides,

from the mid-point of both sides, the trapezium will be

divided into two unequal parts.

5 You must have learned of isosceles triangles, where the two

sides of a triangle are equal and the angle opposite the equal

sides are also equal. In the same way, we have a figure,

which is stated as Isosceles Trapezium, where the two non-

parallel sides are equal and form equal angles at one of the

bases. You can see the example of it, in the third figure given

above.

Properties of a Trapezium

Here, we are going to learn about some more properties of the

trapezium, which is also called as a trapezoid. A trapezium has the

following properties:

1 Like other quadrilaterals, the sum of all the four angles of the

trapezium is equal to 3600

2 A Trapezium has 4 unequal sides

3 A Trapezium has two parallel sides and two non-parallel sides

4 The diagonals of trapezium bisect each other

5 The length of the mid-segment is equal to 1/2 the sum of the

parallel bases, in a trapezium

6 Two pairs of adjacent angles of a trapezium add up to 180

degrees

Area of a Trapezium

Trapezium area can be calculated by using the below formula:

• Area = (1/2) h (AB+CD)

from the mid-point of both sides, the trapezium will be

divided into two unequal parts.

5 You must have learned of isosceles triangles, where the two

sides of a triangle are equal and the angle opposite the equal

sides are also equal. In the same way, we have a figure,

which is stated as Isosceles Trapezium, where the two non-

parallel sides are equal and form equal angles at one of the

bases. You can see the example of it, in the third figure given

above.

Properties of a Trapezium

Here, we are going to learn about some more properties of the

trapezium, which is also called as a trapezoid. A trapezium has the

following properties:

1 Like other quadrilaterals, the sum of all the four angles of the

trapezium is equal to 3600

2 A Trapezium has 4 unequal sides

3 A Trapezium has two parallel sides and two non-parallel sides

4 The diagonals of trapezium bisect each other

5 The length of the mid-segment is equal to 1/2 the sum of the

parallel bases, in a trapezium

6 Two pairs of adjacent angles of a trapezium add up to 180

degrees

Area of a Trapezium

Trapezium area can be calculated by using the below formula:

• Area = (1/2) h (AB+CD)

3.
Perimeter of Trapezium

The perimeter of trapezium formula is given by:

• Perimeter = Sum of all the sides = AB + BC + CD + DA

Derivation of Area of a Trapezium

Following is the derivation for computing the area of the trapezium:

The area of a trapezoid is equal to the sum of the areas of the two

triangles and the area of the rectangle.

We know that

area of trapezoid = area of triangle 1 + area of rectangle + area of

triangle 2.

That means,

A = (ah/2) + b1h + (ch/2)

A = (ah + 2b1h + ch)/2

Simplifying the equation, rearranging the terms, and factoring

result to:

A = h/2[b1 + (a + b1 + h)] ….(i)

If we assume the longer base of the trapezoid be b2, then

b2 = a + b1 + h …..(ii)

Substituting (ii) in equation (i),

A = h/2(b1 + b2)

Therefore the area of a trapezoid with bases b1, b2 and altitude h

A = h/2(b1+b2)

Applications of Trapezium

The concept is a highly used concept in various physics

computations and other mathematical calculations. This is the

basis for obtaining the equations of motion as described in the 9th

CBSE science textbook. The blend of the physics equations and

mathematical calculations is very well explained to clear the level

of understanding of a budding engineering mind.

For the complete understanding and concept clearance from the

BYJU’S one must go through the practice worksheets to check

their hands on the concept and test learning.

Frequently Asked Questions on Area of

The perimeter of trapezium formula is given by:

• Perimeter = Sum of all the sides = AB + BC + CD + DA

Derivation of Area of a Trapezium

Following is the derivation for computing the area of the trapezium:

The area of a trapezoid is equal to the sum of the areas of the two

triangles and the area of the rectangle.

We know that

area of trapezoid = area of triangle 1 + area of rectangle + area of

triangle 2.

That means,

A = (ah/2) + b1h + (ch/2)

A = (ah + 2b1h + ch)/2

Simplifying the equation, rearranging the terms, and factoring

result to:

A = h/2[b1 + (a + b1 + h)] ….(i)

If we assume the longer base of the trapezoid be b2, then

b2 = a + b1 + h …..(ii)

Substituting (ii) in equation (i),

A = h/2(b1 + b2)

Therefore the area of a trapezoid with bases b1, b2 and altitude h

A = h/2(b1+b2)

Applications of Trapezium

The concept is a highly used concept in various physics

computations and other mathematical calculations. This is the

basis for obtaining the equations of motion as described in the 9th

CBSE science textbook. The blend of the physics equations and

mathematical calculations is very well explained to clear the level

of understanding of a budding engineering mind.

For the complete understanding and concept clearance from the

BYJU’S one must go through the practice worksheets to check

their hands on the concept and test learning.

Frequently Asked Questions on Area of

4.
What is the formula for Area of a trapezium?

The formula to calculate area of trapezium is:

Area = ½ x Sum of parallel sides x Distance between the parallel

What are the formulas for Area and Perimeter of

Area of trapezium = ½ x (a+b) x h

Perimeter of trapezium = a+b+c+d

Where a, b, c and d are the length of sides of a trapezium

And h is the distance between the two parallel sides i.e a and b.

What are the properties of Trapezium?

A trapezium has 4 unequal sides: two parallel and two non-parallel

Sum of all interior angles is 360 degrees

Diagonals bisect each other

How do you find the area of Trapezium?

The area of trapezium depends on its parallel sides and distance

between the parallel sides.

If we know the length of parallel sides and the distance between

them, then we can easily find the area of trapezium.

A parallelogram is a two-dimensional geometrical shape, whose

sides are parallel with each other. It is made up of four sides,

where the pair of parallel sides are equal in length. Also, the

opposite angles of a parallelogram are equal to each other. The

area of parallelogram depends on the base and height of it.

In geometry, you must have learned about many 2D shapes and

sizes such as circle, square, rectangle, rhombus, etc. All of these

shapes have a different set of properties. Also, the area and

perimeter formulas of these shapes vary with each other, used to

solve many problems. Let us learn here the definition, formulas

and properties of a parallelogram.

Table of contents:

• Definition

• Formula

• Properties

• Types

• Theorems

• Examples

The formula to calculate area of trapezium is:

Area = ½ x Sum of parallel sides x Distance between the parallel

What are the formulas for Area and Perimeter of

Area of trapezium = ½ x (a+b) x h

Perimeter of trapezium = a+b+c+d

Where a, b, c and d are the length of sides of a trapezium

And h is the distance between the two parallel sides i.e a and b.

What are the properties of Trapezium?

A trapezium has 4 unequal sides: two parallel and two non-parallel

Sum of all interior angles is 360 degrees

Diagonals bisect each other

How do you find the area of Trapezium?

The area of trapezium depends on its parallel sides and distance

between the parallel sides.

If we know the length of parallel sides and the distance between

them, then we can easily find the area of trapezium.

A parallelogram is a two-dimensional geometrical shape, whose

sides are parallel with each other. It is made up of four sides,

where the pair of parallel sides are equal in length. Also, the

opposite angles of a parallelogram are equal to each other. The

area of parallelogram depends on the base and height of it.

In geometry, you must have learned about many 2D shapes and

sizes such as circle, square, rectangle, rhombus, etc. All of these

shapes have a different set of properties. Also, the area and

perimeter formulas of these shapes vary with each other, used to

solve many problems. Let us learn here the definition, formulas

and properties of a parallelogram.

Table of contents:

• Definition

• Formula

• Properties

• Types

• Theorems

• Examples

5.
Parallelogram Definition

A parallelogram is a quadrilateral with two pairs of parallel sides.

The opposite sides of a parallelogram are equal in length and the

opposite angles are equal in measure.

In the figure above, you can see, ABCD is a parallelogram, where

AB//CD and AD//BC.

Also, AB = CD and AD = BC

And, ∠A = ∠C & ∠B = ∠D

Also, read:

5 Diagonal of a Parallelogram Formula

6 Important Questions Class 9 Maths Chapter 9 Areas Parallelograms

Parallelogram Formula

The formula for area and parameter of a parallelogram covered

here in this section. Students can use these formulas and solve

problems based on them.

Area of Parallelogram

Area of a parallelogram is the region occupied by it in a two-

dimensional plane. Below is the formula to find the parallelogram

Area = Base × Height

A parallelogram is a quadrilateral with two pairs of parallel sides.

The opposite sides of a parallelogram are equal in length and the

opposite angles are equal in measure.

In the figure above, you can see, ABCD is a parallelogram, where

AB//CD and AD//BC.

Also, AB = CD and AD = BC

And, ∠A = ∠C & ∠B = ∠D

Also, read:

5 Diagonal of a Parallelogram Formula

6 Important Questions Class 9 Maths Chapter 9 Areas Parallelograms

Parallelogram Formula

The formula for area and parameter of a parallelogram covered

here in this section. Students can use these formulas and solve

problems based on them.

Area of Parallelogram

Area of a parallelogram is the region occupied by it in a two-

dimensional plane. Below is the formula to find the parallelogram

Area = Base × Height

6.
In the above figure, //gramABCD, Area is given by;

Area = a b sin A = b a sin B

where a is the slant length of the side of //gramABCD and b is the

Check here: Area of a Parallelogram Formula

Perimeter of Parallelogram

The perimeter of any shape is the total distance of the covered

around the shape or its total length of any shape. Similarly,

the perimeter of a parallelogram is the total distance of the

boundaries of the parallelogram. To calculate the perimeter value

we have to know the values of its length and breadth. The

parallelogram has its opposite sides equal in length. Therefore, the

formula of the perimeter could be written as;

Perimeter = 2 (a+b)

Where a and b are the length of the equal sides of the

Properties of Parallelogram

If a quadrilateral has a pair of parallel opposite sides, then it’s a

special polygon called Parallelogram. The properties of a

parallelogram are as follows:

6 The opposite sides are congruent.

7 The opposite angles are congruent.

8 The consecutive angles are supplementary.

9 If anyone of the angles is a right angle, then all the other angles

will be right.

10 The two diagonals bisect each other.

11 Each diagonal bisects the parallelogram into two congruent

triangles.

12 The diagonals separate it into congruent.

Area = a b sin A = b a sin B

where a is the slant length of the side of //gramABCD and b is the

Check here: Area of a Parallelogram Formula

Perimeter of Parallelogram

The perimeter of any shape is the total distance of the covered

around the shape or its total length of any shape. Similarly,

the perimeter of a parallelogram is the total distance of the

boundaries of the parallelogram. To calculate the perimeter value

we have to know the values of its length and breadth. The

parallelogram has its opposite sides equal in length. Therefore, the

formula of the perimeter could be written as;

Perimeter = 2 (a+b)

Where a and b are the length of the equal sides of the

Properties of Parallelogram

If a quadrilateral has a pair of parallel opposite sides, then it’s a

special polygon called Parallelogram. The properties of a

parallelogram are as follows:

6 The opposite sides are congruent.

7 The opposite angles are congruent.

8 The consecutive angles are supplementary.

9 If anyone of the angles is a right angle, then all the other angles

will be right.

10 The two diagonals bisect each other.

11 Each diagonal bisects the parallelogram into two congruent

triangles.

12 The diagonals separate it into congruent.

7.
Types of Parallelogram

There are mainly four types of Parallelogram depending on various

factors. The factors which distinguish between all of these different

types of parallelogram are angles, sides etc.

7 In a parallelogram, say PQRS

If PQ = QR = RS = SP are the equal sides, then it’s a

rhombus. All the properties are the same for rhombus

as for parallelogram.

8 Other two special types of a parallelogram are:

• Rectangle

• Square

Is Square a Parallelogram?

Square could be considered as a parallelogram since the opposite

sides are parallel to each other and the diagonals of the square

bisect each other.

Is Rectangle a Parallelogram?

Yes, a rectangle is also a parallelogram, because satisfies the

conditions or meet the properties of parallelogram such as the

opposite sides are parallel and diagonals intersect at 90 degrees.

Parallelogram Theorems

Theorem 1: Parallelograms on the same base and between

the same parallel sides are equal in area.

Proof: Two parallelograms ABCD and ABEF, on the same base

DC and between the same parallel line AB and FC.

To prove that area (ABCD) = area (ABEF).

Parallelogram ABCD and rectangle ABML are on the same base

and between the same parallels AB and LC.

area of parallelogram ABCD = area of parallelogram ABML

We know that area of a rectangle = length x breadth

Therefore, area of parallelogram ABCD = AB x AL

Hence, the area of a parallelogram is the product of any base of it

and the corresponding altitude.

There are mainly four types of Parallelogram depending on various

factors. The factors which distinguish between all of these different

types of parallelogram are angles, sides etc.

7 In a parallelogram, say PQRS

If PQ = QR = RS = SP are the equal sides, then it’s a

rhombus. All the properties are the same for rhombus

as for parallelogram.

8 Other two special types of a parallelogram are:

• Rectangle

• Square

Is Square a Parallelogram?

Square could be considered as a parallelogram since the opposite

sides are parallel to each other and the diagonals of the square

bisect each other.

Is Rectangle a Parallelogram?

Yes, a rectangle is also a parallelogram, because satisfies the

conditions or meet the properties of parallelogram such as the

opposite sides are parallel and diagonals intersect at 90 degrees.

Parallelogram Theorems

Theorem 1: Parallelograms on the same base and between

the same parallel sides are equal in area.

Proof: Two parallelograms ABCD and ABEF, on the same base

DC and between the same parallel line AB and FC.

To prove that area (ABCD) = area (ABEF).

Parallelogram ABCD and rectangle ABML are on the same base

and between the same parallels AB and LC.

area of parallelogram ABCD = area of parallelogram ABML

We know that area of a rectangle = length x breadth

Therefore, area of parallelogram ABCD = AB x AL

Hence, the area of a parallelogram is the product of any base of it

and the corresponding altitude.

8.
In ∆ADF and ∆BCE,

AD=BC (∴ABCD is a parallelogram ∴ AD=BC)

AF=BE (∴ABEF is a parallelogram ∴AF=BE)

∠ADF=∠BCE (Corresponding Angles)

∠AFD=∠BEC (Corresponding Angles)

∠DAF =∠CBE (Angle Sum Property)

∆ADE ≅ ∆BCF (From SAS-rule)

Area(ADF) = Area(BCE) (By congruence area axiom)

Area(ABCD)=Area(ABED) + Area(BCE)

Hence, the area of parallelograms on the same base and between

the same parallel sides is equal.

Corollary: A parallelogram and a rectangle on the same base

and between the same parallels are equal in area.

Proof: Since a rectangle is also a parallelogram so, the result is a

direct consequence of the above theorem.

Theorem: The area of a parallelogram is the product of its

base and the corresponding altitude.

Given: In a parallelogram ABCD, AB is the base.

To prove that Area(||gmABCD) = AB×AL

Construction: Complete the rectangle ALMB by Drawing BM

AD=BC (∴ABCD is a parallelogram ∴ AD=BC)

AF=BE (∴ABEF is a parallelogram ∴AF=BE)

∠ADF=∠BCE (Corresponding Angles)

∠AFD=∠BEC (Corresponding Angles)

∠DAF =∠CBE (Angle Sum Property)

∆ADE ≅ ∆BCF (From SAS-rule)

Area(ADF) = Area(BCE) (By congruence area axiom)

Area(ABCD)=Area(ABED) + Area(BCE)

Hence, the area of parallelograms on the same base and between

the same parallel sides is equal.

Corollary: A parallelogram and a rectangle on the same base

and between the same parallels are equal in area.

Proof: Since a rectangle is also a parallelogram so, the result is a

direct consequence of the above theorem.

Theorem: The area of a parallelogram is the product of its

base and the corresponding altitude.

Given: In a parallelogram ABCD, AB is the base.

To prove that Area(||gmABCD) = AB×AL

Construction: Complete the rectangle ALMB by Drawing BM

9.
perpendicular to CD.

Examples of Parallelogram

Example- Find the area of a parallelogram whose base is 5 cm and

height is 8 cm.

Solution- Given, Base = 5 cm and Height = 8 cm.

We know, Area = Base x Height

Area = 5 × 8

Area = 40 Sq.cm

Example: Find the area of a parallelogram having length of diagonals to

be 10 and 22 cm and an intersecting angle to be 65 degrees.

Solution: We know that the diagonals of a parallelogram bisect each other,

hence the length of half the diagonal will be 5 and 11 cm.

The angle opposite to the side b comes out to be 180 – 65 = 115°

We use the law of cosines to calculate the base of the parallelogram –

b² = 5² + 11² – 2(11)(5)cos(115°)

b² = 25 + 121 – 110(-.422)

b² = 192.48

b = 13.87 cm.

Examples of Parallelogram

Example- Find the area of a parallelogram whose base is 5 cm and

height is 8 cm.

Solution- Given, Base = 5 cm and Height = 8 cm.

We know, Area = Base x Height

Area = 5 × 8

Area = 40 Sq.cm

Example: Find the area of a parallelogram having length of diagonals to

be 10 and 22 cm and an intersecting angle to be 65 degrees.

Solution: We know that the diagonals of a parallelogram bisect each other,

hence the length of half the diagonal will be 5 and 11 cm.

The angle opposite to the side b comes out to be 180 – 65 = 115°

We use the law of cosines to calculate the base of the parallelogram –

b² = 5² + 11² – 2(11)(5)cos(115°)

b² = 25 + 121 – 110(-.422)

b² = 192.48

b = 13.87 cm.

10.
After finding the base we need to calculate the height of the given

To find the height we have to calculate the value of θ, so we use sine law

5/sin(θ) = b/sin(115)

θ = 19.06

Now we extend the base and draw in the height of the figure and denote it as

The right-angled triangle (marked with red line) has the Hypotenuse to be 22

cm and Perpendicular to be h.

sin θ = h/22

h = 7.184 cm

Area = base × height

A = 13.87 × 7.184

A = 99.645 sq.cm

To find the height we have to calculate the value of θ, so we use sine law

5/sin(θ) = b/sin(115)

θ = 19.06

Now we extend the base and draw in the height of the figure and denote it as

The right-angled triangle (marked with red line) has the Hypotenuse to be 22

cm and Perpendicular to be h.

sin θ = h/22

h = 7.184 cm

Area = base × height

A = 13.87 × 7.184

A = 99.645 sq.cm