3 ms·
In a spirit of rough-quantitative reasoning... Because pi is 3, modulo a 5% error we'll so easily ignore, we all know the volume and area of a ball are 1/2 tho
by mncharity 8y ago
In a spirit of rough-quantitative reasoning...
Because pi is 3, modulo a 5% error we'll so easily ignore, we all know the volume and area of a ball are 1/2 those of the box it came in. And the area and perimeter of a circle are 3/4 those of its bounding square. So a unit-wide ball in a box has an area of 3, and its shadow on the bottom of the box has an area of 3/4. QED.
How often does education leave students struggling with half-remembered "was it 4/3 pi r^3... or was it 4 pi r hmm...", and bereft of any clue that the ball's simply half its box. It's like education never uses it to engage with the physical world, so there's no selection pressure for being able to apply it as other than a math tidbit.
- krapht 8y agoWe all know? I think that's a fact that isn't well known.
- dmichulke 8y agoReminds me of the math lectures during university: "Since we all know that ... it is obvious that ..." and I kept thinking "no we don't and no it isn't"
- LanceH 8y agoThose are just stepping stones to "It can be shown" and "Left as an exercise for the reader."
- mncharity 8y agoThank you for posting this comment. I intended "we all know" as a device, an "obvious" counterfactual to emphasize the point, that there's this bizarre gap in what's taught. I wasn't quite happy with it, and later worried it might burn someone ("I guess it's just me that doesn't know that"), but it was late, and I punted. So, my thanks for the save.
- pfortuny 8y agoCareful with areas... Volumes are grasped “intuitevly” but areas are much much harder to “know”.