I am trying to intuitively understand the volume of a cylinder by thinking of the base area times height.
The following video displays the kind of intuition I am looking for, but it is shown for a rectangular prism for which it is easy to see why it works. I wish to apply this same general intuition to volume of a cylinder, but I am stuck since the base is circular and I am struggling with how to think of those unit cubes in the same way.
Say we have a cylinder with height $50$cm and base circle area of $10$cm$^2$. To calculate the volume with this area(by stacking this areas on top of one another until reaching the original height) we need to add a third dimension to the area of circle, i.e. by extending it in height by $1$cm. This will essentially give us a small cylinder whose volume will be the same as the original base area. It is easy to see that from this point we can just stack those small cylinders into the original big one counting how many of those fit and this will be the volume. I am stuck at the part where I create the small cylinder and the whole process feels recursive(proving volume of cylinder with volumes of smaller cylinders). I think I am tunnel visioning here and missing something rather simple, so at this point I am looking for someone to help me believe this in a non-calculus intuitive way.
EDIT: pictures for comment discussion
Best Answer
The volume of any prismatic shape (“stacks” of uniform shape and size) is very simply the sum of the volumes of those little shapes. This works for any (integer, Euclidean) dimension, too. We all know the area of a rectangle: but why is it true? Well, if you subdivide the sides of the rectangle into units (say centimetres) then you can very easily see that the rectangle consists of a certain number of unit centimetre squares, and it all fits, and you can count how many such squares there are by multiplying the side lengths. Then you have $x$ centimetres square. You can’t really do any more proving than that; this is a definition of area - how many unit areas (in this case, unit centimetre squares) cover the shape?
Back to the cylinder now. It really is exactly the same principle. A fundamental definition of volume (in this normal, Euclidean space) would just be: how many unit volumes fill up my shape? And a unit volume can be bootstrapped from the unit area: if you have a square, and add a perpendicular height of a certain length, you can do the subdivision again and see clearly that we have $a\times b\times c$ unit cubes, where $a,b,c$ are side lengths- and then we just define the volume this way. The cylinder consists of a large stack of discs. How thin these discs are and how many discs there are depends on your interpretation, but it is intuitively good to subdivide the cylinder into stacks of unit height. Then the height of the whole cylinder is how many discs there are, and you can say that the volume of the cylinder is height unit discs. But “unit disc” is a clumsy measure of volume - so what’s the volume of a unit disc? Sort of by definition of “unit”, it is just the base area times $1$ units cubed, and again you can’t really push the issue more than that: this is a very natural definition of volume.
Altogether now, we have volume equals $h$ unit discs, and a unit disc has volume $b$, $b$ for base area, $h$ for height, so... $V=b\times h$. The only thing to be wary of is to make sure all your units agree in the same measurement system!
About your issue with the base being circular: this is not actually a problem, so long as you accept the area of a circle. A disc of a unit height is just pulling the base circle upwards, a unit distance, and has volume $1\times b$. If you want an intuition or proof of why the area of a circle is what it is, and why it is even measurable, I’m afraid you probably will have to do some calculus or calculus-type ideas - I believe that’s how the Greeks did it!