# Why is the quantity of a cone one third of the quantity of a cyndrical tube?

The quantity of a cone with elevation $h$ and also distance $r$ is $\frac{1}{3} \pi r^2 h$, which is specifically one 3rd the quantity of the tiniest cyndrical tube that it fits within.

This can be confirmed conveniently by taking into consideration a cone as a solid of revolution, yet I would love to recognize if it can be confirmed or at the very least aesthetic shown without making use of calculus.

You can make use of Pappus's centroid theory as in my answer here, yet it does not give much understanding.

If as opposed to a cyndrical tube and also a cone, you take into consideration a dice and also a square - based pyramid where the "leading" vertex of the pyramid (the one contrary the square base) is changed to be straight over one vertex of the base, you can fit 3 such pyramids with each other to create the full dice. (I've seen this as physical toy/puzzle with 3 pyramidal items and also a cubic container.) This might offer some understanding right into the 1/ 3 "sharp point regulation" (for sharp points with comparable, linearly - relevant cross - areas) that Katie Banks reviewed in her comment.

One can reduce a dice right into 3 pyramids with square bases - - so for such pyramids the quantity is without a doubt 1/ 3 hS. And afterwards one makes use of Cavalieri's principle to confirm that the quantity of any kind of cone is 1/ 3 hS.

An aesthetic demo for the instance of a pyramid with a square base. As Grigory states, Cavalieri's principle can be made use of to get the formula for the quantity of a cone. We simply require the base of the square pyramid to have side size $ r\sqrt\pi$ . Such a pyramid has quantity $\frac13 \cdot h \cdot \pi \cdot r^2. $

Then the location of the base is plainly the very same. The cross - sectional location at range a from the optimal is a straightforward issue of comparable triangulars : The distance of the cone's sample will certainly be $a/h \times r$ . The side size of the square pyramid's sample will certainly be $\frac ah \cdot r\sqrt\pi.$

Once once more, we see that the locations have to be equivalent. So by Cavalieri's concept, the cone and also square pyramid have to have the very same quantity : $ \frac13\cdot h \cdot \pi \cdot r^2$

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