Is there a simple proof for the volume of a cone?


- I think I know the answer!
- That was fast!
- It is not so difficult when you think a bit.
- I am all ears.
- I filled a cone with rice and emptied it into a cylinder. They had the same size circle at the base.
- Did it fill the cylinder?
- It might have, but the cylinder and cone had the same height.

- So?
- I filled the cylinder with exactly 3 cones full of rice.
- So, what does that prove?
- That the volume of a cone is a third of the volume of a cylinder with same radius and height.
- You mean to say that the volume of a cone is \(\frac{\pi r^2 h }{3}\)?
- Exactly.
- Impressive! But I don’t think it counts as a mathematical proof.
- Why not? Are you not convinced?
- Maybe it is not a third, maybe it is 1/2.99!
- That would be ugly!
- So what?
- I live in a beautiful universe.

- That is a strange image! Who made it?
- It is called The Persistence Of Memory by Salvador Dali.

- If you didn’t like my first proof, here is a better one.
- The first was not a proof. But it did suggest a hypothesis.
- What is a hypothesis?
- Something you believe is true, but you haven’t found a proof yet.
- If you find a proof what is it called then?
- A theorem.
- Like the Pythagorean Theorem?
- Exactly. For that one we have a proof.
- OK. Are you ready for my proof number 2?
- Shoot!
- The area of a triangle is \(\frac{base * height}{2}\).

- I totally agree.
- The triangle and the rectangle live in 2 dimensions.
- OK.
- The cone and the cylinder live in 3 dimensions.
- OK.
- The area of the triangle = 1/2 * area of rectangle.
- So?
- So?! Isn’t it obvious? The volume of the cone = 1/3 * the volume of the cylinder.
- Me oh my! You are right. I’ve never thought of that!
- I told you I had a simple proof.
- But, on second thought. Is this really a proof? You are using an analogy that might be right, but it could also be wrong.
- What is an analogy?
- Reasoning by analogy is to use that if two or more things agree with one another in some respects they will probably agree in others.
- But the result could be right?
- Yes, of course, but it could also be wrong.
- Give me an example!
- Listen to this.

Rats are like people in many ways: They have very similar systems of enzymes and hormones, they adapt well to a wide variety of environments, they are omnivores, etc. People carry umbrellas. So, rats carry umbrellas, too. – Source

- But that is nonsense. Rats don’t carry umbrellas!
- Are you sure?

- That is wrong! Mickey Mouse is not a rat.
- We better proceed.
- With what?
- Finding a simple proof for the volume of a cone.
- Oh that. I thought I had given you two proofs already.
- I know.
- So, what is there to do?
- Let’s try to be modest. Whats the volume of these three cylinders inside the cone?

- You mean rectangles, not cylinders!?
- No, they are cylinders. They are seen from the side. That’s why they look like rectangles.
- OK. Now I see.
- So, what is their volume.
- I have no idea. I don’t know how tall they are. Neither do I know their radius.
- OK. That makes sense.
- Thank you.
- Let’s say their height is a quarter of the height of the cone.
- OK.
- The radius question is a bit more interesting.
- You mean more difficult?
- Look at this drawing.

- Look at the triangle in the bottom left corner.
- You mean the one marked h/4 and r/4?
- That’s the one.
- I understand that its height is a quarter of the height of the cone, h/4, but why is its base r/4?
- The ratio between h and r has to be the same as the ratio between the height and base of the triangle.
- And since h/4 : r/4 = h : r everything is in harmony.
- You said it!
- So the radius of the bottom cylinder is \(r-\frac{r}{4}=\frac{3r}{4}\)?
- Correct.
- And the next cylinder has radius \(\frac{r}{4}\) less than that, or \(\frac{2r}{4}\).
- You’re a genius!
- Then I know what the total volume of the three cylinders are!
- Let’s hear it.
- \(\pi (\frac{3r}{4})^2 \frac{h}{4} + \pi (\frac{2r}{4})^2 \frac{h}{4}+ \pi (\frac{r}{4})^2 \frac{h}{4}\)
- Let’s make it simpler.
- Be my guest!
- What about this?
- \(\pi\frac{h}{4}(\frac{9r^2}{16}+\frac{4r^2}{16}+\frac{r^2}{16})\)
- Good! Can you make it even simpler?
- \(\pi r^2 h\frac{9+ 4+1}{64}\)
- \(\frac{9 + 4 + 1}{64}\).
- But that is not a third!
- The idea is to fill the cone completely with cylinders.
- Is that possible?
- That’s a good question! We will know soon.
- How many cylinders do we need?
- Let’s start with n.
- How much is n?
- n is n. Or, to put it differently, n is any number.
- I can dig that. I think…
- The fraction than becomes \(\frac{n^2 + … + 1^2}{(n+1)^3}\).
- It does?!
- In our example we had n = 3 and \(\frac{3^2 + … + 1^2}{4^3} = 0.21875\).
- So, if the patterns holds, we get what you just said. I will check the details later, but it seems reasonable.
- Great.
- But how much is that?
- Let’s ask WolframAlpha.
- Ask who?
- There is a web site called WolframAlpha that can help us.
- How much does it cost to use the site?
- It is free.
- That I like!
- Let’s type (n^2+…+1^2)/(n+1)^3 into the box on the site.

- Look! There is 0.21875 for n=3. Amazing!
- I agree!
- But how do we fill the cone completely with cylinders?
- Let’s try to let n go to infinity.
- What do you mean?
- Let’s see what happens as n grows and grows.
- Can WolframAlpha help us with that too?
- Sure can. We just ask: lim n-> infinity (n^2+…+1^2)/(n+1)^3.
- Can it hear us too?!
- No. What I meant was, “let’s type it into the box” and press Enter.

- It did it! The volume of the cone is a third of the volume of the cylinder. Are you convinced now?
- Actually I am not.
- Maybe after all maths is not for you. Have you tried golf?
- How do we know that the cylinders fill the cone completely?
- Are you blind?! Didn’t you see the third? WolframAlpha got the correct formula. End of discussion.
- Yes, I agree. It found the correct formula, but how do we know that it is the correct one?
- You mean, if we didn’t know already that it was the correct formula?
- Exactly. Like if we were the first ones to find it.
- I see your point.

- You know what, let’s take a break. Maybe you’ll find a way when you wash the dishes.
- Is it related to water?
- I don’t think so, but often when I have a problem I see the solution when I do quite different things.
- How come?
- I don’t know, but I believe that my brain works on the problem unconsciously and suddenly the solution pops up.
- Really?
- Henri Poincare has described a similar experience he had.
- Who is he?
- He was a famous mathematician. Look him up in Wikipedia.

“Just at this time I left Caen, where I was then living, to go on a geologic excursion under the auspices of the school of mines. The changes of travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience’ sake, I verified the result at my leisure.”

- What are Fuchsian functions?
- I have no idea, but that is not important. What is important is that the mind works in mysterious ways.

(To be continued in Milking the solution)

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