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### Course: Get ready for Geometry>Unit 4

Lesson 4: Volume of cylinders, spheres, and cones

# 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?
(62 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.
(59 votes)
• Im still having trouble understanding, too confusing for me
(23 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....
(40 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!!💪
(27 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.
(10 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...
(17 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." -_-
(10 votes)
• Well, you could use it for lots of things! Like it says in the article after the two exercises, you could use it for construction, robotics, drafting, etc. You might not need it, but other people might be interested or need it in the job that they want. I hope that you are satisfied with my answer! :)
(3 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!
(7 votes)
• At about he 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.
(8 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.
(8 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)
• what is the surface area of a cone?
(2 votes)
• Let the radius of the base of the cone be r, slant height be L.
πrL + πr^2
(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.