Hello everyone, welcome back to a new video.
The video we are going to start today is from Geometry
chapter.
And we are doing Mansuration chapter in Geometry.
In Mansuration chapter, the figures we are going to do
today, that would be for 2D figures.
What are 2D figures? Let's understand this first.
2D figures are those figures in which we are given only two
dimensions.
Like you see this smart board, black board or wall of your
house.
In that you will see, one is their length and other is
their breadth or width.
In that, you are measuring that figure on the basis of only
two dimensions.
But when we read 3D, when we talk about 3D, it is when you
go to watch 3D movies.
In which we measure in three dimensions, that is length,
breadth and height.
We will do those figures in the next video.
The purpose of today's video is for only 2D figures.
So, some examples of 2D figures would be coming in your
mind.
Like, the first example is the one which I gave you of
smart board.
So, which figure do we have in smart board?
It is in rectangular shape.
So, the shape would be rectangular, square shape.
Any other example? You have circle, triangle and cases.
Like, there are different types of triangles.
Parallelogram, rhombus, different types of figures of
parallelogram, trapezoid.
Kite, etc. are examples of 2D figures.
So, in today's class, we will see the formulas for area and
perimeter.
In 2D figures, the two things which you subtract from it,
that is area.
Area of any figure means, whatever portion you can see on
this board, covered area.
This white portion.
This area which is being covered on this board is called
area.
We will find the formula for area.
Now, if I just count the boundary of this figure.
Like, if you cannot see the boundary.
If I just see the boundary of this figure.
I can see only the boundary of black colour.
So, if I just measure the boundary of black colour.
So, what is this?
Perimeter.
If we check the boundary of any 2D figure.
So, we will get the perimeter of that figure.
Of that shape.
And whatever area is covered inside it.
That is your area.
Now, let's talk about their unit.
You have studied two things of 2D figures.
That is, one is perimeter.
Perimeter as in, the length of the boundary.
Now, see.
This square figure, rectangular figure, circle, triangle.
You must have made it from anything.
Either you must have made it from pencil.
Or you must have made it from sticks.
Or you must have made it from thread.
Or walls.
You must have made it from any material.
Now, that thing.
That thing is being used by you.
Like, if we are making it from wood.
So, how much wood is being used.
How much length of wood is being used.
To make this shape.
That is your perimeter.
Whatever area is covered inside it.
That is your perimeter.
So, how to find the perimeter?
Whatever area is being used.
To build it.
So, perimeter is your boundary.
Different shapes have different perimeters.
Now, like this rectangular figure.
So, if we add all four sides of it.
So, what will we get?
We will get perimeter.
How much thing is being used by us.
So, unit of perimeter.
It will be in single unit.
Like, it can be centimeter.
It can be meter.
Whatever will be the length of your side.
That unit will be your perimeter.
Like, if it is in 1-2-2 meter.
So, the answer of perimeter will be in same unit.
That is meter.
Now, next thing is your area.
What is area?
Whatever surface is being covered.
Within this figure.
Within this shape.
Whatever shape you are making.
Or rectangle, circle.
Whatever shape you are making.
Whatever region you are getting.
What is that region?
Area.
Area covered by that shape or figure.
Okay.
So, region covered.
Now, what is unit of area?
It is same as perimeter.
Centimeter, meter.
Whatever will be the unit of length.
It will be same as unit of perimeter.
So, it will be unit.
If you are not given.
Then write only unit.
But, what will be in area?
Whatever length unit you have in area.
It will be square.
Means, whatever length of sides you are given.
Suppose, it is given in meter.
So, answer of area will be meter.
In square.
Meter square.
Like, we write square in power 2.
So, write meter square.
Or, if you don't know unit.
Then, we write square.
Sq for square.
And, square unit.
So, now you know.
About unit of area and perimeter.
Now, let's see.
Some formulas of popular figures.
What is their perimeter and area?
First of all, we see.
Most commonly used.
And, even kids know.
Figures which is.
Square.
Rectangle.
And, circle.
We will do of triangle properly.
So, first of all, we are going to do of square.
Area and perimeter.
The first column.
First column is of names.
Of figure names.
Then, second column.
That is for area.
And, third column.
That is of perimeter.
Okay.
You have three columns.
So, first of all, we are doing.
Now, if we make figures of square.
Then, you can easily make figures of square.
Let's see.
If we get square in shapes.
2D figures.
So, tell me which one is square? Now, there is no proper
square in this.
This is.
Right? Let's make square.
Can you see this?
I hope it is visible.
I will increase its width.
Okay.
Now, you will understand.
It will be visible.
Square.
Now, look at its sides.
Sides.
You can't see rectangle.
Consider it as square.
Right? So, what happens in square?
You have four sides.
Four sides make square.
And, what are those four sides?
Equal in length.
Like this side.
This side.
These four sides will be of same length.
Then, it will be square.
And, These two adjacent sides.
The sides which are together.
The sides which are attached.
Right? What do we call them?
Adjacent sides.
Like, look at this side.
This side.
Which side is it attached to?
Which side is it touching?
First, it is touching this side.
And, it is touching this side.
So, this side is adjacent to this side.
And, adjacent to this side.
Let me give their naming.
So that,
Let's suppose, this is square ABCD.
So, how many sides do you have?
Four sides.
Which is side AB?
Side BC?
This, this and DC.
This one and AD.
What will be this?
Four sides.
Now, if this is square,
Then, what will be the length of these four sides?
Equal.
So, length of AB is equal to length of BC is equal to DC is
equal to AD.
Let's suppose, we take that.
Small a and a unit.
Unit can be anything.
Whatever it is.
Centimeter, meter, kilometer.
Right? Whatever the unit is, we have written it as unit for
general.
So, what is a unit?
Length.
Length of each side.
This is also a unit.
Now, what is the second property of square?
The second property of square is that,
The angle between the adjacent sides will be 90 degree.
It should be always 90 degree.
90 degree means,
Adjacent means the same vertex which will have the same
meeting point.
So, AD side is of AB and DC.
DC side is of BC and AD.
One is adjacent side and the other is opposite side.
So, AD side is of AD and DC.
Now, what will be the relation of AD side with BC side?
It will be opposite side.
What will be the opposite side?
So, you have studied two sides.
One is adjacent and the other is opposite side.
I hope this point is clear.
So, what will be the area?
The shaded portion is your area.
So, area will be A into A.
That is side square.
This is what is given.
What will be A square?
Area of any square.
To find the area, you need side.
If you know the length of one side, then it is the same
length of all four sides.
So, multiply them twice.
A into A.
You will get the area.
What will be the perimeter?
Add the length of all sides.
AB plus BC plus CD plus AD.
So, you will have to add A four times.
You will get 4A.
You will have perimeter of 4A.
Similarly, the next figure is rectangle.
Now, rectangle and square are similar.
But there is a slight difference.
Let's go back to shapes.
We are going to shapes.
Let me choose this again.
This is the shape.
Let's fill it with green color.
Let's go.
Can you see?
Instead of green, you can see black.
I don't know why.
You can consider it as black.
This is rectangle.
In rectangle, there are four vertices.
That is points.
Which is A, B, C and D.
This is square.
This is rectangle.
In rectangle, opposite sides are equal.
Opposite sides are equal.
AB equals to DC.
AD equals to BC.
Both are opposite.
AD equals to BC.
Right? Square is perpendicular to adjacent sides.
All four angles are 90 degrees.
Now, you know this.
In square and rectangle, diagonals are formed.
How are diagonals formed?
Join opposite vertices.
Diagonals are formed.
Join opposite vertices.
This is a straight line.
Diagonals are formed.
Diagonals are equal.
Cut diagonals at 90 degrees.
Diagonals are not necessary.
Cut diagonals at 90 degrees.
Diagonals are not necessary.
Cut diagonals at 90 degrees.
Do you understand the difference?
Now, one more thing.
All squares are rectangles.
This is one statement.
All rectangles are squares.
You have two statements.
Statement 1 and statement 2.
Which statement is true?
Are all squares rectangles?
Obviously, all rectangles are equal.
All angles are 90 degrees.
So, all squares are rectangles.
So, statement 1 is true.
Now, look at statement 2.
All rectangles are squares.
Is this statement true?
Now, look at statement 3.
All rectangles are squares.
So, it is not necessary that all sides are equal.
When is it possible?
When all sides are equal.
Then only it is a square.
But, it is not necessary.
So, this statement is not true.
There are many such statements in your question.
Are all rectangles squares?
You should know how to negate these statements.
You should also know which statement is possible.
Now, area of rectangle is length into breadth.
What is length into breadth?
Length into breadth.
And, what is the parameter?
Take L on one side and B on the other side.
So, length is on one side and breadth is on the other side.
So, if this is L, then we will get length on this side.
That will also be L.
If this is B, then we will get length on this side.
That will also be B.
So, length into breadth.
Just like A into A is equal to A square,
Similarly, A is on the other side.
So, length into breadth is equal to L into B.
Now, what is the parameter?
Sum of all sides.
What should we do?
L plus B plus L plus B.
These are all sides.
So, both sides are equal.
So, this is 2B and this is 2L.
Common is 2L plus B.
This is the square parameter.
Next figure is a circle.
Let's make a circle in shapes.
First, let's clean this rectangle.
And, Now, let's make shapes.
Which circle do you see?
Is this a circle or that one?
Can you see the figures?
I am asking two questions.
Is this a circle or that one?
So, we will see that this is a circle.
Circle is oval in shape.
So, this is a circle.
Let's draw a circle.
I hope you can see it.
Let's delete this.
This is a circle.
Let's fill it.
Let's fill the color.
This orange color is a circle.
We make a circle by taking a midpoint.
Which is known as center of the circle.
Let's take a midpoint.
When we make a circle, we keep the compass needle in the
middle.
And then, we stretch the compass as per the radius.
And, we draw a circle with a pencil.
This is the circle.
This is the center of the circle.
You can call it C or O.
And, This is the boundary.
This is the perimeter.
And, You can join this center with any point of the
perimeter.
You can join the perimeter with the center.
You can join it any time.
The length you get is known as radius.
It is known as radius of the circle.
And, when we draw a straight line,
I will explain it with a different color.
If we draw a straight line, what will it be?
It will be the diameter.
The diameter is known as 2 of radius. You should know this.
And, the circumference is known as 2 pi r. 2 pi r is the
circumference where pi is 22 by 7.
Let's take 3.14.
2 pi r is the radius.
The area is known as pi r square.
Sometimes, you may not get r.
Suppose, you get d.
Before getting d, subtract r.
Then, you can put the value in the area and perimeter.
This is the circle.
Now, next.
Next is about Triangles.
Everyone knows about the types of Triangles.
What is a Triangle?
It is a figure made up of three sides.
Let's take three sides.
This is our Triangle.
Let's make it.
Look at this.
This is your Triangle.
Let's fill it with color.
This is a blue Triangle.
This is a Triangle.
It has three sides.
Let's name it.
Side A, Side B, and Side C.
So, we have three sides.
A, B, B, C and A, C.
Now, there are three angles between them.
One angle is A, the other is C, and the third is B.
Now, this angle can be anything.
But what is the relation of these angles?
Write the sum of all three angles.
What should it be equal to?
It should be equal to 180 degrees.
Only then, this Triangle will be formed.
This is the property of a Triangle.
What should be the sum of all three angles? 180 degrees.
Now, you know this.
Now, look at this.
Take any point.
Take C, B, or A.
Suppose, I am taking point A.
From point A, draw a perpendicular line on the opposite
side.
I have drawn a perpendicular line.
Now, we call this perpendicular length as H.
What will we call it?
H.
And, on which you have drawn a perpendicular line?
For example, you have drawn a perpendicular line on A to C,
B.
So, you call C, B as the base.
B.
And, you call this perpendicular length as H, which is
height.
So, from this, you have a formula for a Triangle.
Your formula is for its area, which is 1 by 2 into base
into height.
What is it? 1 by 2 into base into height. Okay? When you
are given the height or the base length of side A, then you
can apply this formula.
But, if you are not given this, the height is not given,
then what is given instead of height? A.
A length is given for A and C.
B length is given for C and B.
And, C length is given for A and B.
Okay? A length, this is B length, and this is your C length.
Okay? C units.
Then, how will you calculate its area? If you are given
only these three.
Okay? Then, the formula for its area is,
First of all, what will you calculate? You will calculate
the sum of these three sides.
And, if you take the sum of these three sides, then you
will get the sum of the boundary, which is equal to our
perimeter of the triangle. Okay? Now, when we take the sum
of these three perimeters, and divide it by 2, then we get
the half of the perimeter.
So, the half of the perimeter, half of A plus B plus C,
which is half of the perimeter.
This will become, you can call it S.
What did we call it?
S.
So, S, you have a new area formula, under root S, S minus
A, means, into S, which is this, before this, minus A
length, then into S minus B, into S minus C.
These four terms are under whose terms? They are under the
under root sign.
One is the term S, then S minus A, then S minus B, and S
minus C.
What is A, B, C?
You have all three sides.
What is S?
You have the half of the perimeter.
What is the perimeter?
Sum of all the three sides.
This is your Heron's formula.
What can you tell about the triangle using Heron's formula?
You can tell the area.
This was the normal triangle.
Anything can happen, whether it is equilateral, isosceles,
scalene, or right triangle.
You can apply this main formula, half base into height, and
if the sides are given, then this formula, if all three
sides are given, then this formula.
But, if you want to remember the inner part, then the
remaining three types are, this is scalene, what is the
scalene triangle?
You have scalene triangle.
Scalene triangle, when all three sides are different, are
unequal.
What is the value of A, B, and C?
They are unequal.
So, this is your scalene triangle.
A not equals to B, not equals to C.
All three are different.
So, you remember this.
Isosceles triangle, means you have two equal sides,
A equals to B, and these two are not equal to C.
They are not equal to third side.
Means, two sides are equal, third side is unequal.
So, remember this, area formula is same, half base into
height, but you will have, perimeter will be different.
It is not different.
You just remember this.
You have to remember this.
This is the main, that there are so many ideas,
classification, but you have to remember this, then you can
put their properties.
For example, if we put here, that isosceles is same, means
A into B, in isosceles.
So, if you put A here, and A here, then both will have same
value.
You just have to do this.
In perimeter also, if you put A here, and A here, then what
will be 2A?
So, here they have written 2A.
Nothing else.
Same, remember the main formula of first triangle. Then,
third is your equilateral triangle, which is all three
sides are equal, and all angles are, all angles are 60
degree.
All angles are 60 degree.
So, sum of all three is 60 plus 60 plus 60 is 180.
So, triangle's property is satisfied.
Now, equilateral triangle, either you do this, half into
base into height, if you have given base and height.
Otherwise, you write, its direct formula, that is under
root 3 by 4, into A square.
Under root 3 by 4 into A square.
What happens in this?
Your height, height, that comes under root 3 by 2.
What happened?
Under root 3 by 2, your height comes.
That is why, if you put A instead of base, because A is
length of side, and if you put under root 3 by 2 instead of
H, then automatically, this formula will be formed.
Under root 3 by 4 into A square.
So, this was for equilateral triangle.
Now, there are more types.
One of the main type of triangle is, right triangle.
Right triangle is, in which one angle is 90 degree.
So, this is 90 degree.
Now, one right angle's property,
Pythagoras theorem, I think I have taught you this in
height and distance.
Pythagoras theorem, if you have A as one side, its length
is A,
BC is one side, its length is B.
So, longest side, length of hypotenuse will be, under root
of these two,
A square plus B square.
This will be your hypotenuse length.
I have taught this in height and distance.
So, this is right triangle.
So, right angle is also a classification.
Next, moving on to,
Rhombus, Parallelogram and Trapezium.
Now, we had done square and rectangle with four sides.
Now, these four are also with four sides.
These are also made with four sides.
But, there are very minute differences.
Like, in rectangle and square, there is a minute difference
that, all its sides will be equal, and only opposite sides
of rectangle will be equal.
If we do rhombus like this, so, look at the figure of
rhombus. If you have a square, you just have to tilt the
square a little bit.
Right? You have to fix one point of the square, and tilt it
a little bit from top.
So, that will be your rhombus.
I will show you how to make it.
This is the figure of rhombus.
Let me fill this and show you.
Rhombus is visible.
Look, all four sides are equal.
This side, this side, this side, this side, this side.
All four sides should have same length.
And, its diagonals should intersect each other at 90
degrees.
Diagonals intersect each other perpendicularly.
And, one more thing, all its angles should not be 90
degrees.
If all its angles are 90 degrees,
I will make a statement
that I
will make a statement that I will make a statement that I
will make a statement that I will sides are parallel sides
and 2 are non-parallel sides now in this what you will do
is the same as you have drawn the height in this take one
point and draw a perpendicular
at the base so you have drawn a perpendicular and you have
given some height of this perpendicular h and you have
given a side of this and you have given b so what will be
the area of this area half into sum of parallel side sum of
parallel side means
a plus b whatever is the parallel side its sum into height
this perpendicular length which you have drawn you have to
multiply it by 2 so half into sum of parallel side into h
this is the formula of area of trapezium now what will be
the perimeter all
four sides are a and b give it c and d sum of all four
sides so you will get perimeter of trapezium so this is
clear to you what is the difference in 4 figure and what is
it so I hope you will clear all the formulas in this video
this was the video of
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