Visualizing the Volume of a Sphere Formula | Deriving the Algebraic Formula With Animations

[pinklady]

I never knew that two identical cones are equal to the sphere of equal height and equal radius. That is very useful to know. As a 3D modeller, I can imagine squashing a cone to make hemisphere.

[jdmc]

I wish all mathematical formulas could be explained in such an intuitive way- Many many thank you's

[TroyaE]

Kyle...

Your animations are excellent.

A useful, and orthodox  method of deriving the formula for volume directly....

Draw your sphere, centre (0, 0).
Allow sphere radius to be r.
Select a value of x to the right of (0, 0).
Erect a perpendicular (perp) of height y.
Rotate that perp about the x axis to form a disc.
Allow that disc to have width dx.
The incremental volume of that disc is its area A = pi*y^2 multiplied by its width dx....
dV = pi*y^2*dx
The perp height y is related to x by the classical equation of a circle...
y^2  + x^2 = r^2
make y the subject...
y^2 = r^2 - x^2
It will follow that...
dV = pi*(r^2 - x^2).dx

To determine the full volume of the sphere, integrate that last equation -r to +r...

V = integral of  pi*(r^2 - x^2).dx  between -r and +r
V = pi*( r^2*x - x^3/3 )

Insert the limits.... -r and +r

V = pi*(  r^3 - r^3/3 - (-r^3 + r^3/3)  ) = pi*( 2*r^3 -(2/3)*r^3 ) 
V = pi*r^3*(2 - 2/3) = pi*r^3*(6/3 - 2/3) = (4/3)*pi*r^3

V = (4/3)*pi*r^3

If, later, you want to get the surface area, simply differentiate the volume function, having identified that... dV = A*dr, so A = dV/dr

dV/dr = A = 4*pi*r^2

Best...

Troya.

[alzobolo]

Dear Sir
God bless you for sharing your knowledge.
I'm a retired engineer trying to fill any gaps in my head.
Lol.
I think that you're the answer.
Smile.
I'm going to subscribe and view all of your work.
Much respect.
Great channel.

[PsyMongazoid]

This is great. Presented beautifully slowly and carefully so you can actually follow it. Love it.

[gaudynguyen]

WOW!! This was so to learn. Thanks for such a clear explanation and clear visuals! My 6th grader was asking me where does the 4/3 come from, and now we both know! :)

[razanshammas]

Oh WOW, this is mind blowing in my opinion. As simply as you put it, it's still amazing. I'll definitely use your method in my lesson planning in the future.
Thank you.

[yourfavoritesport]

This was so simple and well explained. The visuals did great too. Thank you.

[mariaelenarodriguez]

I am 63. I really appreciate this. Thank you.

[SeegalMasterPlayz]

Thanks for making me understand the proof of the formula for the volume of a sphere. This was interesting.

[learnerlearns]

Beautiful, concise, clear presentation. EXCELLENT!

[spiriset]

First time I understand their volume properly thanks and God bless you

[slug]

I always wondered why the volume of the sphere is calculated by that formula, and when I started to see the video where you put water from the cone into the sphere, I immediately understood what you were gonna show me. My head exploded instantly. Now I understand, thanks so much!

[neeraj]

The proof using integration is just as beautiful as this

[harshyofficial]

8 years later, still the best video out there explaining this

[VaneetaMANNERS]

Loved
this video. It was clear and easy to follow. After all these years of teaching, I have not seen such a clear explanation. Thank you!

[amitkumaracharya]

Perfect animation and explanation sir...
You are simply awesome

[idrisayantoye]

Why would anyone dislike this video!
This is awesome. Great job (now I don't have to memorize)

[MatthewCahn]

Great explanation. But how did we find that the volume of a sphere = the volume of two cones with the same height and radeii as the sphere?

[Sameer.K]

Simply amazing. I wish our world has more teachers like you. God bless you