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.)

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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
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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).

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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.

http://2.bp.blogspot.com/__wa77chrZVg/SuRA4fj-l8I/AAAAAAAADHM/quRNFMVeHmk/s400/Chou_pei.jpg

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
    image
  • The “large” size may not look much bigger but its volume can in fact be.
    image
  • 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.)

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    Small circle has radius, the distance from the center to the edge, called “r”. Smaller circle has an area, space inside,...
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