Why is the volume one third of that? I mean, where's the fault in my logic?

Question

The volume of a cuboid is $l \times b \times h$. That is, it is equal to base area times height. I think it means that the base is added up height times, it becomes volume (It makes sense if we think about it)

And if we think about cylinder, the above mentioned logic still holds. The volume of cylinder is $\pi r^2 \times h$ . Which is same as saying the area of bottom circle (i.e. $\pi r^2$) times the height $h$.

But when it comes to a cone , it doesn't makes sense. In above examples 2-d shapes have been collected one upon another, a specific number of times(h). In a cone its kinda same except that now a triangle isn't collected one upon another (that would make a prism), but collected in a circular way. We can imagine a cone to be group of right triangles with their axis being one of the sides (except hypotenuse, that would make a compound shape of two cones ). We can also imagine it this way, A right triangle is rotated with one of the sides being axis.

So now the volume should be, by above mentioned logic, the area of repeated shape times number of shapes. And there are 2pi x r triangles because, well, it is rotated that much number of times. Visualize it in your mind, it'll become clear. Hope you get the idea.

Now, Volume of cone should be area of the triangle times the circumference of the base

=> $1/2 \times b \times h \times 2\pi r$

Now the base of triangle is same as the radius of the base and the 2s cancel each other out we are left with

$\pi r^2 h$

But the volume of cone if one third of that value. My question is Why is that?

Answer 1

This could be explained well with integrals, but i'll try to give an explanation that builds on your reasoning.
The thing is, you can't exactly obtain a volume from sheets, like the first pic suggests.
what you can obtain a volume from is only other volumes.
So, the first pic only makes sense if we think of the sheets as "small volumes"; cuboids with volume l x b x h with h very small (infinitesimal, in technical terms).
Just look at a book: it's made of sheets of paper, and it looks tridimensional enough. and so are the sheets of paper from which it's made, though their height is very small. (an infinitesimal height would be even smaller, but let's not worry too much, the image offered by the pages of a book is good enough for what comes next) Ok, so, the cylinder works the same way: small cilinders stacked upon each other. now. if we expand this way of seeing things to the cone... what do we need to stack to obtain a cone?
We wound need to stack... like, tiny slices of cone.
I mean, let's think of a cake: we could see it as a lot of tiny slices of cake "stacked" together around the center.
If we have a cake in the shape of a cone, we can still make "slices": solids with four faces, two of which identical triangles (the sides of the slices). The "top" of the slice is on the surface of the cone, and the base on the base of the cone.
So, what we need to stack to obtain a cone is actually these sort of slices, if we intend to use a rotated triangle to decompose the cone...
The reasoning you propose actually uses triangle-shaped "sheets", not slices, so you're basically trying to build a cone-shaped cake by stacking triangle-shaped "sheets"... and that's kind of impossible to do.
EDIT: i made a picture of dubious quality that sums up what i was trying to say and adds one thing or two.

Answer 2

Rotation of a shape $S$ which has an area $A$ can result in different volumes depending on the shape of $S$, even if the area of $S$ stays constant. As an example, consider a rectangle with sides of length $1$ and $2$. If you rotate it around the shorter side, you get a cylinder of volume $\pi r^2 h = \pi(2)^2(1) = 4\pi$. But, if you rotate it around the longer side, you get a cylinder of volume $\pi r^2 h = \pi (1)^2(2) = 2\pi$.

Some points on the shape are further from the axis of rotation than others, which causes this effect. Even if you have the exact same shape, but increase the distance to the axis of rotation, the volume of the resultant shape will increase.

That is why you can not know the volume of a rotation of a shape just by the area of the shape.

This answer may explain it better.