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 is indeed a matter of being 3D.

Instead of a cylinder, let’s first visualize a 2D version: a triangle. The area of a triangle is b*h/2: half the product of the base and the height; in other words, half the area of its enclosing rectangle. To picture this, imagine taking slices of the triangle. At the bottom, the length is b; a third of the way up, it’s 2/3 of b; at the top, it’s zero. On average, it’s half the length of the base, so multiply that by the height.

Now when we move to 3D, the volume of a prism or a cylinder is b*h: area of the base, times the height. Since every slice is just like the base, this makes sense. But when you draw a cone in the cylinder (or a pyramid inside a prism), what is the average area of a slice? When you are halfway up, that slice is not half the area of the base; it’s a quarter of the area. So you’re not taking the “average” of *x*, as you did with a triangle; you’re taking the average of *x*^(2) instead. This average, letting *x* go from 0 to 1, turns out to be 1/3 (the exact value requires calculus to explain).

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.

What you say is only true for a slice that passes through the center of the cone perpendicular to the base. What does the next slice look like? Progress the slice (in parallel) towards one edge of the cone? Do you still have the non-cone area = cone area in that slice? In fact, do you expect to see a triangle?

It’s a Calculus thing. Going from a square to a cube is going from X^(2) to X^(3) . One of the rules of calculus is that exponents become coefficients. so X^(2) becomes 2X and X^(3) becomes 3X^(2) .

If you want to think about it more spacially, remember that area goes up expodentially with radius. A circle with a radius of 1″ has about 3.142 square inches while a circle with a radius of 2 has about 12.5 square inches of radius.