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### Course: Geometry (all content)>Unit 8

Lesson 4: Volume of cones, cylinders, and spheres

# 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?
• 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.
• Im still having trouble understanding, too confusing for me
• 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

I hope this helps someone....
• Can someone explain the easier way to do this? The video is way to confusing and complex. Thanks
• 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!!💪
• 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.
• 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...
• "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." -_-
• 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! :)
(1 vote)
• how can we find the surface area of cone?
• 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!
• At about he says multiplying by 3 is the same as dividing by 1/3. How is that true?
• 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.
• Why is a cone's volume 1/3 of a cylinder's?
• 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. :)