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### Course: High school geometry > Unit 9

Lesson 3: Volume and surface area# Volume of a cone

The formula for the volume of a cone is V=1/3hπr². Learn how to use this formula to solve an example problem. Created by Sal Khan.

## Want to join the conversation?

- why 1/3 and not 1/2? seemed like the area of a triangle which is half of the square or rectangle built upon its base. why 1/3?(56 votes)
- remember: in a triangle, the base is straight. In a cone it is a circle. The cone's volume is 1/3 the volume of the cylinder with the same base.(48 votes)

- Im still having trouble understanding, too confusing for me(22 votes)
- Ok...first of all khan academy went the hard way for solving this problem...there is an easier way which i am about to show...

Question : 131cm^3 = 1/3*5*πr^2

r=?

Formula : V = 1/3*Hπ*R^2

Work :

131cm^3 = 1/3 * 5 * π * r^2

*(around)(estimate) after calculating 1/3*5 π

131cm^3 = 5.2 * r^2

131cm^3 = 5.2 * r^2

/ / *diving 5.2 on both sides....

5.2 5.2

131/5.2 = r^2

25.19 = r^2 (around)(estimate)*divided 131/5.2...

*square root both sides....

√(25.19) = √(r^2)

5 (estimate) = r

radius = 5

I hope this helps someone....(33 votes)

- What if we have the radius and volume but not the height.

I need to know how to find the height. Don't forget this kind of thing may take a long time.(9 votes)- To solve for the height we need to isolate variable 'h' in V=1/3hπr².

V = 1/3hπr²

3V = hπr²(Multiply by 3 to remove the fraction)

3V/πr² = h(Dividing both sides by 'πr²' isolates 'h')

With this new formula(3V/πr² = h), you can substitute the valve of the volume and the radius and solve for the height.

V=131

h=approx. 5

3(131)/(π x 5²) = h = approx. 5

When we solve for the height we get 5 back which is the height of the cone...(16 votes)

- Can someone explain the easier way to do this? The video is way to confusing and complex. Thanks(7 votes)
- Sure thing Frankie! I hope mine isn't too confusing though!😄

You know, I'll just keep it simple!

Let's start!😎

-Cones are like pyramids, except that they're with a circular base. Maybe that's what makes you confused, but I've got a trick that'll hopefully help you!💡

-So if you make an experiment, by bringing an empty cone, and a cylinder filled with water (they must be the same base length)... pour the cylinder's water in the cone, 2 3rds would be left, so the cone only takes a third of the cylinder's volume. Thus, The cone's formula is the cylinder's multiplied by 1/3 so it would be written like this: V= 1/3 πr^2h OR V= πr^2h/3 (since multiplying 1/3 is the same as dividing by 3).🧐

Hope that was useful!!

#YouKhanLearnAnything!!💪(20 votes)

- It is confusing. I know that this is the same formula but it might be a little more clear. Formula: volume = (1/3) · π · r^2 · h

Hope this helps. :)(9 votes)- Thanks!

Some simple steps I created to find the volume of a cone:

*This is without a calculator in terms of π.*

1. Square the radius

2. Multiply the result by the height

3. Divide by 3

^- Keep as a fraction if the result isn't a

whole number

4. Plop a π at the end

Example:

Radius = 3

Height = 6

1. Square the radius

3^2 (or 3 x 3 ) = 9

2. Multiply the result by the height

9 x 6 = 54

3. Divide by 3

54/3 = 18

Since the result is a whole number, we do not need to represent it with 54/3.

4. Plop a π at the end

18 becomes 18π

There's your answer!

This is especially helpful on the practice sheets.

I have only tested this when the radius and height are both whole numbers so it might not work on decimals / fractions.(11 votes)

- how can we find the surface area of cone?(5 votes)
- Good question!

The surface area of a cone is pi*r^2 + pi*rL, where r is the radius of the base and L is the**slant**height.

(Note that L is**not**the same as the perpendicular height, h, that appears in the formula for the volume of a cone.)

Have a blessed, wonderful day!(6 votes)

- At about4:20he says multiplying by 3 is the same as dividing by 1/3. How is that true?(3 votes)
- When you divide by a fraction, you are actually multiplying by its reciprocal. That is: a/b, when b is a fraction, is equal to a * 1/b. You can try it yourself with something like 6/3.(7 votes)

- "how are you gonna use this in a real life situation?"

*pulls out a calculator in an icecream shop to figure out how much ice cream you can fit in a waffle cone.*

"you can fit aproxamitly 7.4598074358049579845790583oz in a waffle cone." -_-(6 votes) - Why is a cone's volume 1/3 of a cylinder's?(4 votes)
- Because it's just a rounded version in 3D of a regular triangle.

think about it a tringle splits a a square in half.

It's the same thing!

A cone is half a cylinder! like an hourglass!

I hope this helped. :)(3 votes)

- How are these formulas made? What is the logic behind them?How do you consider them in each equation?(2 votes)
- Many geometric area and volume formulae require calculus (particularly integral calculus) to derive. Some don't, however. In fact, cones, circles, and even spheres have area/volume formulae that were discovered without modern calculus, though the techniques used closely resembled calculus.(7 votes)

## Video transcript

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.