If there is a cylinder, and a cone inside the cylinder, shouldn’t one “empty space” on either side of the cone, be half the volume of the cone? It has the same height, radius, etc. Of a cone. So shouldn’t the equation for half a cone be 1/2 ( 1/3πr^2). Then meaning that both the halves equal one, and that a cylinder is equal to two cones instead of three??? I just don’t understand conceptually how they aren’t equal. Where is that third cone going?? Does it have something to do with it being 3D??? I’m not stupid I swear I just can’t visualize it😭
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It works with any “pyramid” of arbitrary shaped base. The base could be a circle (cone) or a triangle or a square or a rectangle or whatever. As long as there in a single point up top, the area is the (1/3)(area of base)(height).
It has to do with dividing it into cross-sectional areas that scale from 0 (at the point) to the full area of the base. What you are doing is summing areas (which are proportional to the “square” of the distance from the point) from the zero point to the base. When you sum squares, it turns out that the leading term is n^3 / 3. There’s other terms (for whole numbers, those terms would be n^2 / 2 and n/6 ) but to get an accurate volume you need to take very thin slices. Those lower terms become vanishingly small as you take thinner slices.
I guess that’s not really ELI5, but that’s the best that I can do. When you sum up progressively increasing squares you get a third of a cube.
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