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).
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