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Without using calculus, it’s actually pretty mysterious where that 1/3 factor comes from. I found this somewhat shocking—- usually there is some easy (but perha

Without using calculus, it’s actually pretty mysterious where that 1/3 factor comes from. I found this somewhat shocking—- usually there is some easy (but perhaps clever or elegant) framing whereby the answer just “falls out,” but not in this case!

jaclaz 3y ago
There is the demonstration analyzing a frustum and then extending its smaller radius until it is the same as the base one (i.e. stretching the frustum until it becomes a cylinder): https://web.maths.unsw.edu.au/~mikeh/webpapers/paper47.pdf https://web.maths.unsw.edu.au/~mikeh/webpapers/paper47.pdf This does not use calculus at all but still needs some time to digest, the idea is that there must be a coefficient c that multiplied by the base area of the cone A and by its height gives the volume (V=cAh) and that if we express the volume of the cylynder with the same approach we get V=3cAh.
- pharrington 3y ago
  You said that demo doesn't use calculus at all, and yet I see a sum of infinitesimal divisions of a surface area. I know an integral when I see one.
  - jaclaz 3y ago
    Look only at page 1, where the volume of the cone is calculated. The sum of surfaces is on page 2 about the surface of a sphere.
    - pharrington 3y ago
      My bad!
- deleted 3y ago
  [deleted]
- Solvency 3y ago
  I love how this document just randomly throws in "e" which mysteriously means "extra" with absolutely zero further explanation or rationale. This is why mathematicians drive me insane. There is zero boilerplate for explaining where these variables come from and why. But, cool. Let's just roll with this "extra".
  - consilient 3y ago
    What explanation is there to give? It's just the ordinary meaning of the word: when you cut out a shape you end up with some extra material left over.
  - zem 3y ago
    a frustum is the base of a cone when you slice off a conical cap, so the remaining height of the cone from which the frustum was sliced is the "extra".

jaclaz 3y ago

There is the demonstration analyzing a frustum and then extending its smaller radius until it is the same as the base one (i.e. stretching the frustum until it becomes a cylinder): https://web.maths.unsw.edu.au/~mikeh/webpapers/paper47.pdf https://web.maths.unsw.edu.au/~mikeh/webpapers/paper47.pdf This does not use calculus at all but still needs some time to digest, the idea is that there must be a coefficient c that multiplied by the base area of the cone A and by its height gives the volume (V=cAh) and that if we express the volume of the cylynder with the same approach we get V=3cAh.

pharrington 3y ago
You said that demo doesn't use calculus at all, and yet I see a sum of infinitesimal divisions of a surface area. I know an integral when I see one.
- jaclaz 3y ago
  Look only at page 1, where the volume of the cone is calculated. The sum of surfaces is on page 2 about the surface of a sphere.
  - pharrington 3y ago
    My bad!
deleted 3y ago
[deleted]
Solvency 3y ago
I love how this document just randomly throws in "e" which mysteriously means "extra" with absolutely zero further explanation or rationale. This is why mathematicians drive me insane. There is zero boilerplate for explaining where these variables come from and why. But, cool. Let's just roll with this "extra".
- consilient 3y ago
  What explanation is there to give? It's just the ordinary meaning of the word: when you cut out a shape you end up with some extra material left over.
- zem 3y ago
  a frustum is the base of a cone when you slice off a conical cap, so the remaining height of the cone from which the frustum was sliced is the "extra".