# Your Toilet Paper is Lying to You

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# Picture this —

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.

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.

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.

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

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:

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)

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:

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.

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).

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.

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.

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…

… to get this equation:

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.

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.

So the toilet paper would be halfway done somewhere around the dotted line —

If you don’t want to go through all the math yourself, you can check out this Javascript code I made to fit the equation for your bathroom needs. Try it out yourself here.

May you go forward with this knowledge as a more enlightened individual.