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## Class 10 (Foundation)

### Course: Class 10 (Foundation)>Unit 7

Lesson 2: Cone and sphere

# 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....
• 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...
• 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!!💪
• 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. :)
• Thanks!

Some simple steps I created to find the volume of a cone:
*This is without a calculator in terms of π.*

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:
Height = 6

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

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.
• 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.
• How are these formulas made? What is the logic behind them?How do you consider them in each equation?
• 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.
• 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. :)
• How do I find the surface area of a cone?