Post #2712461
2026-02-28 22:37 UTC
@sibrosan@mastodon.social @divbyzero@mathstodon.xyz
4. Notice that s is contained in S. If we remove s from S, the remaining solid is the part of the shell between P and the corresponding piece of the inner sphere. It has the volume
volume(S) - volume(s)
= volume(S) * (1 - (r/R)^3)
= area(P) * R / 3 * (1 - (r/R)^3)
Replies (1)
-
@hcschuetz@mastodon.social 2026-02-28 22:37
@sibrosan@mastodon.social @divbyzero@mathstodon.xyz The precise formula here is actually not important. The important observation is that this volume depends only on the *area* of P. It depends neither on the shape of P nor on its position on the outer sphere. So if you do steps 1 to 4 for two slices P and P' of the outer sphere with the same area, the resulting "rings" will have the same volume. So, does this convince you?