Proof for Volume of a Sphere (without Calculus)

In this video, I go through a proof for the formula of the volume of sphere without the use of calculus.
I've also built an interactive Geogebra applet to see the cross-section in 3D:
sites.google.com/view/matholi...

[@MassMultiplayer]

for some reason i understood everything it make sence. but i also instnatly saw i would have taken my whole life to even consider this. for some reason the water 3d displacement is genius , love it subed to yor chanel

[@ChenHongming]

It is customary now that when you understand the water displacement concept to run around the streets naked shouting Eureka! 🤪

[@302ci1968]

Very very interesting, thank you !

[@culater]

Very nice 👍

[@blackchicken2243]

Very cool video

[@ChenHongming]

Thank you! 😃

[@chasejohnson8326]

very cool :)

[@ChenHongming]

Thank you for your kind words!

[@Filipnalepa]

For sake of being complete-how do we show that a volume of a cone is r²*h/3?

[@ChenHongming]

That would need another (and longer) video. But I would approach it from proving a square-based pyramid of unit height and base (I made a quick animation of that here: instagram.com/p/CiXFgQapGh4/). Then through scaling and shearing transformations, prove that it works for triangular base. I will also need Cavalieri's principle to explain why shearing preserves the volume. Then using triangular base to prove for polygons then showing that if it works for polygons, it must also work for circles (i.e cone = circle-based pyramid) and circles are just polygons with 'infinite' sides.

[@adb012]

@@ChenHongming ... "If it works for polygons then it works for circles" is fine. But the problem is that it works for squares and triangles doesn't immediately mean that it works for all polygons and hence for circles. I prefer the following method: First, you can calculate the area of the circle with the most-calculus-like thing that the Greek did without actually using formal calculus, which is to slice the circle in very thin "pizza" slices, re-arrange the slices alternatively right-side-up and upside-down to form a rectangle of height r and base circumference/2, and finally define pi as the ratio between the circumference and the diameter to get to pi*r^2 for the are of the circle. Then, obtain the area of a squared-base pyramid in function of the area of the base and the height (as you did in your animation): Vol of squared-base pyramid Vp = 1/3*Ap*h where Ap is the area of the base of the square-pyramid. Then observe that a cone would have the same cross-section area at any height than a square-based pyramid at the same height, and hence would have the same volume. The assertation of equal cross-sectional areas at equal heights can be done observing that a) the cross-section of the area of the square-based pyramid is a^2 (where a is the side of the square of the cross section at the selected height) so area of the cross section is proportional to a^2, b) The cross-sectional area of the cone is pi*r^2, where r is the radius of the cross-sectional circle at height h, so the area of the cross section is proportional to r^2, c) that a and r change linearly from zero at the top to the max value at the bottom and d) that the areas are the same at the bottom. Finally, Cavalieri's principle implies that 2 solids that have the same cross-sectional area at any height also have the same volume. So if the pyramid and the cone have the same volume, and the same base area, then the volume formula in function of the base area also needs to be the same, so Vc = 1/3*Ac*h, were Ac is the area of the base of the cone pi*r^2, so Vc = 1/3*pi*r^2*h. QED.

[@maciej12345678]

9:02 because yellow triangle has 45 degrees

[@GolfUB]

in 2:42 why did you changed the volume formula of a cylinder to be pi * r^2 * 2r instead of pi * r^2 * h?

[@ChenHongming]

I am considering a cylinder that has the same height as the sphere.

[@miguelmouta5372]

👍

[@Dineshkumar-by9px]

Awasome

[@_learn_for_life_]

Ouhhhhhhhhhh 😁

[@SimchaWaldman]

Here is another method I discovered: kzbin.info/www/bejne/j5rTdJxjdpprr7M

[@alexglaras5090]

Mans pfp is korosensei lol

[@ChenHongming]

And was drawn using Geometer's SketchPad.

[@edutech3270]

Bro, without knowing that volume of sphere is (4/3)pi.r^2, how did you state that theorum??You are saying that volume of sphere is (4/3)pi.r^2 and then proving it??

[@ChenHongming]

Using a displacement can, you can work that out experimentally first like the story of archimedes and the king’s crown. This gives you a hypothesis that you can then prove using various methods.