## Thursday, June 23, 2011

### Toilet Roll

Do you ever wake up in the morning, go to the bathroom, and wonder "how much smaller would the toilet roll be if it didn't have that hole in the middle?".

Yeah, me too.

As it turns out, it's not that hard to estimate. First of all, we note that the roll is near enough to a regular cylinder that we can simplify the problem to two dimensions. We're then just dealing with areas of circles.

Of course, you could just re-roll an actual toilet roll by hand to find out. But that would be silly.

Here's a diagram
The left one is the roll as it is, the green bit of the right is the roll as it would be if it didn't have the hole in the middle (r2 being the new radius), and dr is the change in radius size - what we want to work out.

The point here is that the areas of matching colour in each diagram are the same size. So we can equate the formulas for the (green) areas and use a bit of algebra to find an equation for dr.

The full derivation is left as an exercise for the reader.

The resulting equation looks like this
And plugging in values measured from an actual toilet roll we get,
So the answer turns out to be a decrease of about half a centimeter, or of ~9.5%. Which isn't that much - about 1/5th the radius of the hole.

So now you know.

Oatzy.

[Before you say anything, read what it says at the top of the blog]