All language subtitles for 967.English

Do volume and capacity

mean the same thing?

The answer is no.

There's a very subtle difference between them.

In order to understand that,

let me give you a simple example.

Let us take a wooden box

open from the top.

It has some thickness ofcourse.

The space occupied by the box

is its volume.

And the space available for you

to keep something inside

is its capacity.

If I keep increasing

the thickness of the box,

you can easily see

that the space inside it

keeps decreasing.

Or in other words

its capacity is decreasing.

But it's volume remains the same.

If I keep increasing the thickness,

the box will become fully solid

and there'll not be

any space available inside it.

This means it has zero capacity.

But still the volume

remains the same.

This means that any object

will have volume but it may

or may not have a definite capacity.

So volume is the space

occupied by an object,

whereas capacity refers to the

ability of the object

to contain something.

Let's quickly summarize

what we've seen so far.

The shapes that we've seen

and what the surface areas

and volumes for

each of these shapes are. Right.

The entire chapter,

all of surface area and volume

is nothing but this 1 table

that we'll now see.

Let's start with the

first shape that we saw.

Cuboid with dimensions

length*breath*height.

How will you find

the total surface area

of this cuboid?

Let’s first find

the area of the flat faces.

The cuboid has only flat faces.

Let me open up the cuboid.

What do you see here?

You see 6 rectangles.

2 of them

have area length*breadth.

2 of them have area

breadth*height.

2 of them have area

length* height. Right.

So the surface area,

the flat surface area

is going to be

2(length*breadth+breadth*height+

length*height).

The curves surface area is zero.

The cuboid does not have

any curved surfaces.

So the total surface area

will be same as

the flat surface area

that we saw.

Coming to the volume,

for the volume

think of 1 rectangular plate

with length=L

and breadth=B.

Now stack up

H such plates. Right.

So what is a total volume

you’ll get?

1 plate had an area

length * breadth.

H such plates.

So length * breadth * height.

That is nothing but the

volume of your cuboid.

Next figure that we saw was a cylinder.

Let's take a cylinder with

radius r and height H.

This is the cylinder that you have.

It has 2 flat faces. Right.

If we open this out

this is nothing but a circle

and this is a circle. Right.

The area will be

πr² and πr²

Total flat surface area

will be 2πr²

Now the remaining portion

of the cylinder is

nothing but the curved portion.

Let me cut it here

and open it up.

What do you see?

Nothing but a rectangle.

This length

is nothing but 2πr. Correct.

Because it was a circle originally.

It was a circle. So it's 2πr

When we open it up it remains the same.

So this is 2πr

and this H.

Area of this rectangle will be

2πrh.

This is nothing but the

curved surface area

of the cylinder. Right.

Total surface area,

just add the 2.

2πr²+2πrh. Right.

So you’ll get 2πr(r+H).

That's the total surface area.

For the volume,

what is a cylinder really?

If you take 1 circular plate.

Right. Of radius H

and stack up

H such plates,

you'll get a cylinder. Right.

You'll get a cylinder.

So volume is nothing but

this area πr²*H because you have

H such plates. Right.

That's the volume of your cylinder.

πr²H.

The next figure

is a cone.

Let's take a cone

with radius of base r,

height H

and slant height L. Right.

Slant height L.

The flat surface

that you see here is

this circle at the bottom.

If you remove this,

this is nothing but a circle.

Area πr².

That's the flat surface area.

The remaining portion of the cone

we have left is

the curve surface area.

If I now cut it from here

and open it out,

what you'll have

is the sector of a circle,

with this radius = L

and this length

of the sector = πr. Right.

If you add these up

the total surface area will be

πr*r+L.

Now volume of a cone.

If you take a cylinder

we already know the

volume of a cylinder.

Now take a cone

with same radius of base

as cylinder and same height

as cylinder.

If we pour water

from the cone to the cylinder,

you will see that

you'll require to pour it

3 times to fill the cylinder.

Which tells you,

volume of the cone is

1/3 the volume of the cylinder.

Right.

So volume of cone will be

1/3πr²H.

Let's take a sphere now.

If you look at this

the sphere has

no flat surfaces. Right.

So flat surface area

will be 0.

It only has curved surfaces.

If you measure the

curve surface area of a sphere,

it will be the same as

the surface area of

4 circles

of the same radius. Right.

So if we have

this sphere of radius r,

we'll take 4 circles

of the same radius r,

the area of these 4

will be the same as

area of the sphere.

Right.

So it's nothing but 4πr²

Now the total surface area

will remain the same 4πr²

because it had no flat faces.

If you take the volume,

the volume of this sphere

is nothing but 4/3πr³ Right.

So the volume of this sphere

will be 4/3πr³.

That's it.

The entire chapter is complete.

All that you need to remember

is this one table. Right.

And once you know the visualisation

you don't even need to

learn this table. Right.

Just remember the visualisation

and you'll automatically know

what the formula is.

That's it.

The chapter is really over.

Just these 5 minutes

and the chapter is over. Right.

This is all that

you need to know.

Problems is nothing about

putting numbers into these formulae,

in some cases combining

2 of these kinds of objects

to find area.

That's it.

There is nothing else to this.

Right.

Just these 5 minutes

and the chapter is done.

After that the only thing

you have to take care of is

not making calculation mistakes.

That's it.

5 minutes and your

chapter is complete. Right.

One of the biggest chapters

and 5 minutes and it's complete.

Right.

So always visualize in geometry.

More importantly in 3 dimensional geometry

it's very important to visualize.

The moment you visualize

you don't need to memorize

any of this. Right.

If you think of the visualization,

you'll automatically know

what the formulae need to be.