toilet paper calculus
The other day I started realized that toilet paper is pretty non-linear. (As I’m not the first person to point out.) The last part of the roll disappears a lot quicker than the start because each sheet makes up more and more of a warp around the roll.
I decided to do some calculations and make it real by measuring (using my awesome new Mitutoyo calipers I love) a roll of 350 sheet two-ply toilet paper I have here.
Conceptually one can look at it as a series of concentric shells of radius Rn and length 2*pi*Rn. A roll with no center tube looks like:
L = pi*R^2/T + pi*R
where L is the length of paper, R is the radius, T is the paper thickness.