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
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.
@@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.
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??
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.
A sphere has no other dimensions than its own diameter. A cylinder is not dependent on its own diameter it also has a length that can vary. How did they come to this conclusion that a sphere is 2/3 of a cylinders volume? What cylinder? This doesn’t make any sence. If you fill a sphere with water you can see for yourself that the diameter of the sphere is all the information we need to calculate the volume. If the diameter (inner measurements) e.g is 15cm you can easily transform it to centilitres by multiplying it with 10. Så 15*10 = 150 centilitres. Which is the same as 1,5 liters. Why do we need cones and pi and make this unnecessary confusing when all we want is to know the amount of empty room inside of it.
From a pragmatic engineering perspective you are correct, you can just measure it to be around 2/3 of a cylinder. But from a mathematical perspective, we want to know (a) it is exactly 2/3? or could it possibly be 2.013/3? and (b) how can we prove it? We can't just experimentally induce it from measurements.
