Prove: Volume of a Sphere (Disk Method)

Let's prove the formula for the volume of a sphere using disk method! Each step is carefully explained with an intuitive appeal--if you are looking for a comprehensive overview of the proof, this is the video for you!

Your support is truly a huge encouragement.
Please take a second to subscribe in order to send us your valuable support and receive notifications for new videos!

Every subscriber and like are immensely appreciated.

[HonorableScaevola]

So glad you made this. I wanted to solve using this method but I couldn't set it up properly. Thanks!

[nullplan]

Another proof without calculus:

Take a hemisphere with radius R. A plain that is parallel to the base of the hemisphere will have a circular cross-section with the hemisphere. If h is the distance from the plain to the base, then the radius will be sqrt(R² - h²), so the circles will have area pi r² = pi (R² - h²).

Now take a cylinder of radius and height R and cut out a cone such that the base of the cone is the top of the cylinder. A plain that is parallel to the base of the cylinder will have a ring shaped cross section, with the outer radius R and the inner radius being h (since the cone is straight, and turns out to be a rotated version of y=x). So the area of the ring is pi R² - pi h² = pi(R² - h²).

So, since it is therefore possible to slide a plain over both of these figures such that their cross-section with the plain is always equal, they must have the same volume. Volume of the cylinder is pi R³, and the volume of the cone is 1/3 pi R³. Thus the volume of the hemisphere is 2/3 pi R³, and, by symmetry, the volume of the whole sphere is double that, so 4/3 pi R³.

[arya_moghadam]

Thank you so much, this was so well explained 👍🏼

[ahmedghouri]

Thanks for this .
You’re a great man 👍

[mattpachec]

thank you so much!! officially my favorite explanation.

[matthewmazurek]

Now just gotta prove the volume of a disk... 😬