So I understand that the volume formula of a cone is: $\frac{1}{3}\pi r^2h$, and when I read about how to derive this formula, it makes sense to me. Funny thing is, I'm stuck on why it ISN'T $\pi r^2h$ when I think about deriving the volume formula in a different way.
Here's what I mean. Now we're told that to figure out the volume formula for something like a cylinder or cube, we simply multiply the area of the base by the height, and this makes intuitive sense. In the case of a cone, what if we took a triangle with base r and height h, and rotated it around some axis, producing a cone. When I think of it this way, it seems reasonable (in my mind) to calculate the volume of the produced cone by taking the area of the triangle, and multiplying it by the circumference of the circle that is the cone's base. So if the area of our triangle is: $\frac12rh$ and the circumference of the generated cone's circle-base is: $2\pi r$, then the volume should be: $\pi r^2h$. This derivation seems intuitively correct to me.
Now I know it isn't correct, and I almost feel silly asking, but I just can't see WHY it's not correct. What's the flaw in the reasoning that the volume of a cone can be derived by multiplying the area of the triangle by the circumference of its circe-base?
Best Answer
What you need to look at is the second theorem of Pappus. When you rotate things in a circle, you have to account for the difference in the distance traveled by the point furthest from the axis (the bottom corner of our triangle) which goes the full $2\pi$ around. However, the vertical side doesn't actually move, so it doesn't get the full $2\pi$.
Pappus' theorem says you can use the radius of the centroid as your revolution radius, and it just so happens the the centroid of this triangle is a distance $r/3$ from the axis.
Another good read is this.