Volume of a cone

Let's think a little bit
about the volume of a cone.

So a cone would have
a circular base,

or I guess depends on
how you want to draw it.

If you think of like a
conical hat of some kind,

it would have a
circle as a base.

And it would come to some point.

So it looks something like that.

You could consider this to
be a cone, just like that.

Or you could make it
upside down if you're

thinking of an ice cream cone.

So it might look
something like that.

That's the top of it.

And then it comes
down like this.

This also is those
disposable cups of water

you might see at
some water coolers.

And the important
things that we need

to think about when we want to
know what the volume of a cone

is we definitely want to
know the radius of the base.

So that's the
radius of the base.

Or here is the radius
of the top part.

You definitely want
to know that radius.

And you want to know
the height of the cone.

So let's call that h.

I'll write over here.

You could call this
distance right over here h.

And the formula for the
volume of a cone-- and it's

interesting, because it's close
to the formula for the volume

of a cylinder in a
very clean way, which

is somewhat surprising.

And that's what's
neat about a lot

of this three-dimensional
geometry

is that it's not as messy as
you would think it would be.

It is the area of the base.

Well, what's the
area of the base?

The area of the base is
going to be pi r squared.

It's going to be pi r
squared times the height.

And if you just multiplied
the height times pi r squared,

that would give you the volume
of an entire cylinder that

looks something like that.

So this would give
you this entire volume

of the figure that
looks like this, where

its center of the top is
the tip right over here.

So if I just left
it as pi r squared

h or h times pi r
squared, it's the volume

of this entire can,
this entire cylinder.

But if you just want the
cone, it's 1/3 of that.

It is 1/3 of that.

And that's what
I mean when I say

it's surprisingly clean that
this cone right over here

is 1/3 the volume of this
cylinder that is essentially--

you could think of this
cylinder as bounding around it.

Or if you wanted
to rewrite this,

you could write this as 1/3
times pi or pi/3 times hr

squared.

However you want to view it.

The easy way I remember it?

For me, the volume of a
cylinder is very intuitive.

You take the area of the base.

And then you multiply
that times the height.

And so the volume of a
cone is just 1/3 of that.

It's just 1/3 the volume
of the bounding cylinder

is one way to think about it.

But let's just apply
these numbers, just

to make sure that it
makes sense to us.

So let's say that this is
some type of a conical glass,

the types that you might
see at the water cooler.

And let's say that
we're told that it

holds 131 cubic
centimeters of water.

And let's say that we're
told that its height right

over here-- I want to do
that in a different color.

We're told that the height of
this cone is 5 centimeters.

And so given that,
what is roughly

the radius of the
top of this glass?

Let's just say to the
nearest 10th of a centimeter.

Well, we just once again
have to apply the formula.

The volume, which is
131 cubic centimeters,

is going to be equal
to 1/3 times pi

times the height, which is 5
centimeters, times the radius

squared.

If we wanted to solve
for the radius squared,

we could just divide both
sides by all of this business.

And we would get
radius squared is

equal to 131 centimeters
to the third--

or 131 cubic centimeters,
I should say.

You divide by 1/3.

That's the same thing
as multiplying by 3.

And then, of course, you're
going to divide by pi.

And you're going to
divide by 5 centimeters.

Now let's see if we
can clean this up.

Centimeters will cancel out
with one of these centimeters.

So you'll just be left
with square centimeters

only in the numerator.

And then to solve
for r, we could

take the square
root of both sides.

So we could say that
r is going to be

equal to the square root of--
3 times 131 is 393 over 5 pi.

So that's this part
right over here.

Once again, remember
we can treat units just

like algebraic quantities.

The square root of
centimeters squared-- well,

that's just going to be
centimeters, which is nice,

because we want our
units in centimeters.

So let's get our calculator
out to actually calculate

this messy expression.

Turn it on.

Let's see.

Square root of 393 divided by 5
times pi is equal to 5-- well,

it's pretty close.

So to the nearest, it's
pretty much 5 centimeters.

So our radius is approximately
equal to 5 centimeters,

at least in this example.