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
This is also how I first really understood √2, now my favourite number.