# How to intuitively see that area of base times the height gives volume of cylinder

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

• Just approximate the base with a polygon (having a large number of sides) and thus the cylinder with a prism. Commented Sep 2, 2021 at 9:53

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!

• The answer is perfect. Just one thing regarding the circle/cylinder. When we think about area/volume in terms of numbers of unit squares/cubes covering it how do we handle, in the case of the circle/cylinder, those squares/cubes that are not perfect? Such as those on the edges of the circle/cylinder (that have a side curved). Once again, intuitively if possible. Commented Sep 2, 2021 at 12:30
• @MichaelMunta So this is what I meant by my final paragraph. Let’s assume that the base area of the circle is true and a given. Then its covering of unit squares (which we are assuming exists, for now) becomes a covering of unit cubes when you add a unit height to make the disc. Then volume equals $1\times \rm base$. But why is there a covering of unit squares for the area of a circle? Well, this does become a calculus problem, but intuitively a circle is approximately a polygon when you push the number of sides as large as it can go. Commented Sep 2, 2021 at 12:33
• @MichaelMunta (cont.) then a circle does have an area, and there does exist a way to cover it with squares - you just need the squares to become extremely (“limitingly”) small, and you need a lot (infinitely many) of them. But if you trust in the area of a circle, then you can trust in the volume of a disc! Commented Sep 2, 2021 at 12:35
• is it safe to think about those imperfect unit cubes as parts of a unit cube? So in the case of that unit disc we have normal unit cubes around the middle of the disc and we have "parts of unit cubes" around the edges. This makes it easier to for me to accept that by adding height to a circle we get a unit disc which we multiply to get full volume. Say calculus told me exactly which fraction of a unit cube those imperfect cubes are, then I would count those, count the normal ones and I would get the volume of a unit disc. Is this a valid way of thinking? Commented Sep 15, 2021 at 15:05
• It’s not quite that we get imperfect cubes. The full blown calculus answer would tell you that we take ordinary cubes, or indeed any partition of 3D space, and find the area as the partitions tend in size toward zero. The intuitive response might be: “itty-bitty” cubes around the edges and cubes with a more standard size in the centre @MichaelMunta Commented Sep 15, 2021 at 15:07

If you understand multivariable geometry, this will be pretty simple. An example of cylinder in $$\mathbb{R}^3$$ is a circle in the $$xy$$-plane projected through the $$z$$ axis by some height.

Personally, I don't think that the linked video provides a good foundation for developing intuition around the volume of a cylinder.

The video presupposes that the area of a cube whose sides all $$= 1$$ is given by $$1^3.$$ Then, it proceeds to establish the volume of the lowest level of $$12$$ cubes as $$12$$, and then it provides intuition to establish that the volume of the (rectangular) prism is
length $$\times$$ width $$\times$$ height.

There is no corresponding intermediate result that may be used for a similar foundation of the volume of a cylinder. That is, you would be forced to assume that the volume of a cylinder of height $$= 1$$ will be given by $$\pi \times$$ the square of the radius. This begs the question of how you know the formula for the volume of a cylinder whose height $$= 1$$.

Personally, although I was exposed to the formula for the volume of a cylinder prior to my taking Calculus, I was never taught any justification for the formula until I started Calculus. Then, the formula grew to seem sensible, because you took the cross-sectional area times the height.

Personally, I regard the consideration of cross-sectional area to be somewhat advanced for pre-Calculus students.

This is how I understood volume of cylinder.

1)Take a circle ,we know that it's area is πr².

2)One important fact is that the thickness of the the circle is too small to be compared with its radius.

3)Now take n numer of those circles,stack them up to a height h.

4)Now we know that the general formula for Volume is Area *Height .

1. We know the area ,multiplying this by the height ,gives us the volume of the cylinder formed by the stack of circles as πr²h.

Hope this helps.