Sep 17, 2022

5 min read

Your Toilet Paper is Lying to You

Picture this —

Man sitting on toilet — Why is the bathroom rug there? Why is he using a flip phone? Why am I writing this article in the first place?

You are in the bathroom, sitting on the toilet right now. You look to your side. You see it. The toilet paper. Resting there on the holder, acting all innocent.

picture of toilet paper — I looked up “ominous toilet paper” but nothing could top this.

You notice the toilet paper is about halfway gone. “Time to buy some more, you think.” BUT IS IT?

For you see, the toilet paper doesn’t want to be replaced. It wants you to wait until the very last second before getting more. It is lying to you, and you don’t even know it.

picture of toilet paper laughing maniacally — Pretty happy with my font choice on this one.

But luckily, if you take that toilet paper, and look through the cardboard, you will see there is a light at the end of the tunnel.

a boy looking through toilet paper rolls — I would like to thank the human race for making this thing of beauty.

See, when you look at a toilet paper roll, you probably are thinking this is the halfway point —

a diagram of a toilet paper roll — Look into the depths of the toilet paper, and you will see yourself as you truly are.

It makes sense. Halfway down the toilet paper roll, it sure seems half of the toilet paper has been used. BUT IT’S A LIE.

The truth is, finding the actual center point is going to require some serious skills. Skills that you can only learn after a lifetime of intensive pooping. But I didn’t have time for that, so I just made an equation to figure it out myself. So without further ado(uce), here is The Answer:

overall toilet paper equation — Please keep reading. I know you don’t want to, but please.

Wait, what?

Yeah, we need to break that down, no laxative needed.

So let’s start by defining our variables:

a = diameter of entire toilet paper
b = diameter of inner tube
y = 1/amount you are looking for left (so since we are looking for half of the toilet paper used, y is going to be 2)

diagram of toilet paper explaining equation variables — I used y as the variable because the letter itself is a question.

Now, that we have our variables, we can start breaking down the equation.

The first equation we need is finding the area of a circle from the diameter. This is the below:

diameter to area equation — “Why not just use radius instead, so you don’t need to divide by two?” You try finding the center point of a toilet paper roll sometime and get back to me about that.

But toilet paper isn’t a normal circle. There is a second big circle smack dab in the middle of it. Therefore, we need to figure out the area of the circle that actually has toilet paper, which is just the bigger circle area minus the little circle area.

toilet paper area equation — This also is the equation for a donut.

We now need to find half of that area to find the halfway point of toilet paper use (this is where the 2 in the denominator is taking the place of the y variable).

half of toilet paper area equation — I feel like I’m insulting your intelligence with this equation explainer

But we aren’t trying to find the area. We are trying to find the diameter of the toilet paper when it reaches it’s halfway point. This is represented by x.

finding halfway point equation — I feel like it’s Thanksgiving with all this Pi.

I then move the formula around so I can get variable x (the length of the diameter when half of the toilet paper is gone) all alone.

simplified finding halfway point equation to isolate the diameter variable — What an obnoxiously large root. What does this equation think it is, a turnip?

And this is a pretty good equation to find the point in the toilet paper roll in which half the toilet paper has been used. If you want in in terms of a percentage instead of the length of diameter, you can combine the above equation with the below equation…

percentage equation — What a breath of fresh air

… to get this equation:

percentage and diameter length equation combined — Just remind yourself that letters can’t hurt you.

And voila! You have defeated the scheming toilet paper with the power of math, casting a fatal blow to its trickery, and can now use this knowledge to have a more enlightened bathroom experience.

boy dressed as a toilet paper knight — Become the toilet paper knight!

After taking a random sample of different toilet papers, you can see that the typical halfway point on a given toilet paper roll is ~63% between the edge of the cardboard and the full toilet paper roll.