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Calculate the Thickness of Toilet Paper with Math

Correctly hung toilet paper.  Source: Wikimedia Commons.
Correctly hung toilet paper. Source: Wikimedia Commons.

If you're like me, you probably like to do math problems in your head while you sit on the throne. One of the easiest calculations you can do is figure the thickness of your toilet paper using nothing but measurements from a fresh roll and the length of the paper as given on the package. If the diameter of the entire roll is D, the diameter of the inner tube T, and the paper length L, then the thickness of the toilet paper is given by the formula

paper thickness = (π/4)(D^2 - T^2)/L

If the paper is n-ply, the then thickness of an individual ply is given by

ply thickness = (π/4)(D^2 - T^2)/(n*L)

Here we explain the derivation of the formula and show an example computation.


Formula Explanation

If a roll of toilet paper is unfurled and viewed from the side, the profile is that of an extremely elongated rectangle. The area of that profile is the length of the roll times the height of the paper, i.e., the thickness. If we call the length L and the thickness X, then the area is given by the expression

area = LX

What if we roll the paper around a tube? Still looking at it from the side, the shape will change (from a rectangle to an annulus), but the area will remain the same because we are not adding or subtracting any amount of paper. If the rolled toilet paper has an outer diameter of D and a tube diameter of T, then the area is given by the expression

area = (π/4)(D^2 - T^2)

Since both area expressions are equal, we have

XL = (π/4)(D^2 - T^2)

Solving for X, the thickness, gives us

X = (π/4)(D^2 - T^2)/L


Example

I computed the thickness of my own toilet paper to discover just how little or how much my family's collective cheeks are being pampered during throne time. The first thing I had to figure out is how long a roll is. My brand of toilet paper did not make this easy for me! The package says that each "square" is 9.9 cm long by 10.1 cm wide, and that each roll contains 200 squares. Thus, the length of one roll is

L = 200*9.9 = 1980 cm.

I must find the length of the roll using the number of squares per roll and the dimensions of each square. (Source: Me)
I must find the length of the roll using the number of squares per roll and the dimensions of each square. (Source: Me)
Measure the diameter of the whole roll.
Measure the diameter of the whole roll.
Measure the diameter of the tube.
Measure the diameter of the tube.

Next I measured the diameter of a fresh roll. Unfortunately, the rolls get a little squished and are not perfectly circular, so you may have to re-squish them into shape. Using my best estimate I got a diameter measurement of 11.4 cm.

You can more accurately estimate the diameter of the roll by measuring its circumference and the dividing by π. I applied this method and got a circumference of 36.1 cm, which means the diameter is 36.1/π = 11.5 cm. I split the difference and used D = 11.45 cm.

To measure the diameter of the tube, estimate where the center is and measure across. I got T = 4.2 cm.

Now plug all the variables into the toilet paper thickness equation:

(π/4)(11.45^2 - 4.2^2)/1980

= 0.045 cm

= 0.45 mm

My toilet paper is 3-ply, so each ply has a thickness of 0.045/3 = 0.015 cm, or 0.15 mm. How thick is your toilet paper?

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Comments 10 comments

Rochelle Frank profile image

Rochelle Frank 3 years ago from California Gold Country

Wow.. Calculus, I am higher-math impaired, and have to admit that I have never been mentally motivated to do math problems in my head... in the head.. or that particular venue.

However, since your hub was based on something that is almost universally known (TP), I was tempted to read or at least skim your article. I certainly made me think of things in a different manner--- like the very elongated rectangle. I can hardly get my head around it.

God job, I think. :)


Rochelle Frank profile image

Rochelle Frank 3 years ago from California Gold Country

Of course-- I meant, GOOD job.


CWanamaker profile image

CWanamaker 3 years ago from Arizona

Haha pretty awesome. This was certainly a good read - I learned something new today!


eugbug profile image

eugbug 3 years ago from Ireland

I worked out the solution by first finding the average circumference which is π(D+T)/2.

Dividing the total length L by this figure gives the number of concentric circles which is 2L/(D+T). This is an average figure so it must be doubled giving 4L(D+T). The distance between the core and the perimeter is D-T so dividing this by the number of circles gives (π/4L)( D² - T² ).

Your solution is more elegant though!.

What happens if the paper is embossed.....


calculus-geometry profile image

calculus-geometry 3 years ago from Germany Author

Very nice, eugbub. I like that you also obtained the number of layers wrapped around the tube. When the toilet paper is embossed or quilted you might get an overestimate unless the paper is wrapped tightly. I was going to write this article a few weeks ago, but our toilet paper stock at the time was some cheap stuff with too much air between the layers.


Beth Eaglescliffe profile image

Beth Eaglescliffe 3 years ago from UK

This has made me think about toilet rolls in a whole new way! Thanks for writing a great article.


PoeticFailosophy profile image

PoeticFailosophy 3 years ago from Cuzco, Peroo

I use an old phonebook, will this calculation still work?


calculus-geometry profile image

calculus-geometry 3 years ago from Germany Author

Depends on how you use the phonebook. When you're on the john, do you call random people and ask if they have any tp, and if so, how thick it is?


Buildreps profile image

Buildreps 22 months ago from Europe

Very nice, calculus-geometry. I didn't do this with toilet paper but with Aluminum foil. I don't want to get to serious about this, but isn't the thickness of the toilet paper also depending of the relief patterns as well?


calculus-geometry profile image

calculus-geometry 22 months ago from Germany Author

Yeah, if you have the quilted kind it's harder to get it accurate. They even have rolls without the cardboard tubes nowadays, so there's another monkey wrench. Thanks for the comment!

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