Posted on Friday, 3 May 2013

OK, not every day. But whenever I shop for packaged retail goods like a coffee or in the grocers.

The Pythagorean theorem demonstrates that a slightly larger circle has twice as much area as a slightly smaller circle.

Pythagorean Theorem  This is how I first really understood the Pythagorean Theorem.  The outer circle looks just a little bit larger than the inner circle. But actually, its area is twice as large.  Kind of like the difference between medium and large soda cups, or how a tiny house still requires kind of a lot of timber, for how much air it encloses. If you buy a slightly wider pizza or cake it will serve proportionally more people; and if an inverse-square force (sound, radio power, light brightness) expands a little bit more it will lose a lot of its energy.  Ideas involved here:  scaling properties of squared quantities(gravitational force, skin, paint, loudness, brightness)  circumcircle & incircle  2  This is also how I first really understood 2, now my favourite number.

(Since the diagonal of that square is √2 long relative to the "1" of the interior radius=leg of the right triangle. So the outer radius=hypotenuse=√2, and √2 squared is 2.)


And some of us know from Volume Integrals in calculus class that a cylinder's volume = circle area × height — and something like a sausage with a fat middle, or a cup with a wider mouth than base, can be thought of as a “stack” of circle areas
or in the case of a tapered glass, a “rectangle minus triangle” (when the circle is collapsed so just looking at base-versus-height “camera straight ahead on the table” view).


The shell-or-washer-method volume integral lessons were, I think, supposed to teach about symbolic manipulation, but I got a sense of what shapes turn out to be big or small volume as well.

By integrating dheight sized slices of circles that make up a larger 3-D shape, I can apply the inverse-square lesson of the Pythagorean theorem to how real-life “cylinders” or “cylinder-like things” will compare in volume.

  • A regulation Ultimate Frisbee can hold 6 beers. (It’s flat/short, but really wide)
    File:Frisbee Catch- Fcb981.jpg
  • The “large” size may not look much bigger but its volume can in fact be.
  • Starbucks keeps the base of their Large cups small, I think, to make the large size look noticeably larger (since we apparently perceive the height difference better than the circle difference). (Maybe also so they fit in cup holders in cars.)

162 notes

  1. viraltruth reblogged this from isomorphismes
  2. ajora reblogged this from thescienceofreality
  3. thekingoflegoland reblogged this from thescienceofreality
  4. cosmicunicorn reblogged this from thescienceofreality
  5. electroncloud reblogged this from thescienceofreality
  6. science-sexual reblogged this from protomeathean
  7. cornbreab reblogged this from forrestsaabcehmu
  8. forrestsaabcehmu reblogged this from cornbreab and added:
    Small circle has radius, the distance from the center to the edge, called “r”. Smaller circle has an area, space inside,...
  9. frischerluft reblogged this from thescienceofreality
  10. twerkswithwolves reblogged this from thescienceofreality
  11. rneta-rnaus reblogged this from thescienceofreality
  12. punchlineloser reblogged this from scienceing
  13. thepumpkinpyg reblogged this from thescienceofreality
  14. thedjin reblogged this from thescienceofreality
  15. nerdandtall reblogged this from thescienceofreality
  16. poutine-putain reblogged this from scienceing
  17. cumberston-hiddlesbatch reblogged this from thescienceofreality
  18. polkmn7749 reblogged this from thescienceofreality
  19. protomeathean reblogged this from scienceing