Does the diameter of a toilet roll or kitchen paper roll decrease exponentially?

There were a few answers, including “It’s a logarithmic spiral”, “It’s hyperbolic” and “It’s linear”. There were some complicated equations for Archimedean spirals. Here’s what I think.

We can reasonably say that the area of the edge of the paper is the same if it’s laid out flat, or if it’s rolled up in a spiral. Near enough, anyway. So:

wnl = πr²

Where w is the width of the paper, n is the number of sheets, l is the length of a sheet, and r is the radius of the roll.

We can solve that to get r = sqrt(wnl/π).

and thence

diameter = 2 * sqrt(wnl/π)

No, I am not going to implement proper maths notation in this blog. Not without a better reason than this.

Anyway the point is, I reckon the diameter is roughly double the square root of some constant times the number of sheets. I did a model in Blender to illustrate:

Near enough, considering it’s only supposed to be an approximation? Of course if we’re talking about the decrease of the diameter of a toilet roll, that graph will be flipped on the x axis and will stop when it hits the diameter of the cardboard tube.

All of this assumes that paper usage is linear over time, which is probably true-ish although in real life it’ll be subject to biological and behavioural noise that isn’t really any of my business.