Timeline for Explaining why volume of cone is a third of cylinder
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 1, 2022 at 13:07 | comment | added | Steven Gubkin | @fedja I actually teach the argument your way to future educators (since I agree: it is easier to visualize), but it complicates the $\frac{1}{3}$ with an extra factor of $\frac{1}{2}$. If you do not an excellent grasp of multiplication of fractions this can trip you up. | |
| Jun 1, 2022 at 0:47 | comment | added | fedja | I find it a bit easier to imagine $6$ pyramids of height $a/2$ with bases on the cube faces and the apex at the cube center, but that is certainly a matter of taste. | |
| May 5, 2022 at 12:11 | comment | added | Steven Gubkin | @Arthur Unfortunately, mathematical induction is actually a form of deductive reasoning. en.wikipedia.org/wiki/Inductive_reasoning | |
| May 5, 2022 at 9:26 | comment | added | Arthur | "[Math] is a deductive art." I would actually go so far as to say it is an inductive art. | |
| May 4, 2022 at 20:15 | vote | accept | Bravo | ||
| May 4, 2022 at 11:53 | comment | added | Steven Gubkin | @WillOrrick This description also gives a probabilistic interpretation of the formula $\int_0^1 x^n \textrm{d}x = \frac{1}{n+1}$: mathoverflow.net/a/114739/1106 | |
| May 3, 2022 at 17:51 | comment | added | Will Orrick | @MichałMiśkiewicz Yes. Or, for each of the $n$ coordinate axes, you get an $n$-pyramid whose altitude is the interval from 0 to 1 along that axis and whose cross sections are $(n-1)$-dimensional cubes given by $0\le x_j \le x_i$, where $x_i$ is the coordinate in the chosen direction and $x_j$ for $1\le j\le n$, $j\ne i$ are the $n-1$ other coordinates. More simply stated, the $i^\text{th}$ pyramid is characterized by the condition that $x_i$ is the coordinate with greatest value. | |
| May 3, 2022 at 17:21 | comment | added | Michał Miśkiewicz | Thanks to both of you! @WillOrrick's pictures suggested to me that in $n$ dimensions you can choose two opposite vertices $A=(0,\ldots,0)$ and $B=(1,\ldots,1)$ (for example), and for each "$n$-dimensional side" adjacent to $B$ (there are $n$ of them) take the convex hull of $A$ and that side. | |
| May 3, 2022 at 15:37 | comment | added | quarague | @MichałMiśkiewicz It does generalize in the sense that in 4 dimensions the constant is 1/4, in 5 it is 1/5th and so on. I think you can cut an n-dimensional hypercube into n hyperpyramids suitably analogous to the 3d example but I would have to draw and think about it some more. | |
| May 3, 2022 at 2:09 | comment | added | Brendan W. Sullivan | (+1) Great answer, and especially for the idea of a "theory of the volume of cones." At first, I thought, "Yeah, that's a good conversation to have with students if they're learning volume formulas in this way." But also, this might be my go-to answer for "How is math different from science?" I think going into great detail to describe the "more and more exacting experimental set ups" required to develop the great "theory of the volume of cones" will really illustrate the point :-) | |
| May 2, 2022 at 13:35 | comment | added | Will Orrick | By the way, "of course" doesn't mean I saw this immediately the first time I thought about it. It means that after I understood it, I slapped my forehead and said, "of course!" | |
| May 2, 2022 at 13:19 | comment | added | Will Orrick | @MichałMiśkiewicz Of course. I made some pictures that, I think, give the general idea. The link is here. I'm not so good with graphics so you may have to fiddle a bit with the animation controls to get a good view. You can stop/restart, speed up/slow down, rotate, and resize the images. | |
| May 2, 2022 at 13:06 | comment | added | Michał Miśkiewicz | This approach shows the origin of $1/3$ at the very beginning, which I think is great - a student may be used to the $1/2$ in $2D$ are formulas, so that's the primary point to address, in my opinion. Does the 3-pyramids-construction generalize to higher dimensions? | |
| May 2, 2022 at 11:18 | history | answered | Steven Gubkin | CC BY-SA 4.0 |