The Toilet Paper Theorem: The Mathematics behind An Ordinary Roll
While watching a movie with my family, I came across a Charmin ultra soft commercial that claimed their newest toilet paper roll, The Charmin Ultra Soft Super Mega Roll Toilet Paper, had six times as many toilet paper sheets compared to a regular toilet paper roll. The Charmin ultra soft roll was indeed thicker because it had more sheets, but I wanted to know how much thicker the Charmin ultra soft roll was compared to the regular toilet paper roll based on their claim. In other words, how much longer is the radius of the Charmin roll to the the radius of the regular toilet paper roll? Furthermore, I wanted to explore if there was a mathematical explanation to the thickness of the toilet paper roll.
What Factors Influence the Thickness of a Toilet Paper Roll?
There are multiple things to consider when measuring how much thicker one roll would be compared to another. You would need to account for the thickness and dimensions of the toilet paper sheet along with the total number of toilet paper sheets and the radius of the cardboard roll in the middle. However, the most important thing you would need to consider is the changing circumference. Every time the toilet paper is wrapped around the cardboard roll, it gets thicker. This means that the circumference of the next wrap will be larger than the wrap before it. This also means the amount of toilet paper sheets used for each wrap will increase because more toilet paper is needed to cover a larger circumference. We would need to derive the number of sheets as a function of the radius, which would tell us how many sheets of toilet paper are needed to wrap a certain radius. The derivation for this theorem will be based on Figure 1 which depicts the central cardboard roll, represented by the inner brown circle, and multiple wraps around it, illustrated by each black circle. The line labeled rc represents the radius of the cardboard roll and the line labeled rf represents the radius of the entire toilet paper roll.
Finding the Equation for the Length of the Toilet Paper
This derivation will begin with the combined length (L) of the toilet paper sheets. This length (L) is a sum of all the circumferences of the toilet paper wrapped around the central cardboard roll. We know that the radius of each circle is changing with respect to the thickness (p) of a single sheet of toilet paper and the number of wraps (n). So we can map our length as an equation seen below.
This equation can be expressed in sigma notation as seen below.
Finding the Amount of Wraps in a Particular Roll
From these equations we can’t determine how many wraps (n) are in a particular toilet paper roll. The way to find the number of wraps is to consider how the radius of the roll is changing as the number of wraps is increasing. Figure 2 shows how the number of wraps multiplied by the thickness gives you the distance between the largest radius and the innermost radius.
This will yield the equation below.
Next we can isolate n to get the number of wraps in a toilet paper roll to satisfy the equation below.
Now that we have a defined number of wraps (n), we can determine the length (L) of the toilet paper, by plugging the equation for the number of wraps into the length equation. This will yield the equation seen below.
Using the Sum of an Arithmetic Series Equation to Traverse the Length
Instead of adding up each and every term, we can do this in a much simpler way. Since this equation represents a sum of an arithmetic series, we can plug this into the equation below where Sn represents the sum of all the terms, a1 represents the first term in the series, an represents the last term in the series, and ns represents the number of terms being added.
In our case, the total number of terms (ns)can be represented through expression below. Since n starts from zero and not one in the length equation, we have to add one.
Using our length equation in a sigma notation, the first term (a1) is 2π(rc) and the last term (an) is represented through the expression below, which can be simplified to 2π(rf).
Using the sum of an arithmetic series equation and the expressions above, we can find the length to follow the equation below.
We can simplify the equation above to get the length of the toilet paper to follow the equation below.
My original goal was to find the amount of sheets using the radii. So we end up with the equation below where S represents the amount of sheets, rf represents the radius of the entire toilet paper, rc represents the radius of the central cardboard roll, p represents the thickness of a single sheet of toilet paper, and w represents the width of a single sheet of toilet paper.
This equation represents the number of sheets in terms of constants. If we were to represent the amount of sheets as a function of the radius (r), we would write the equation below, which tells us the amount of toilet paper sheets at a particular radius in the toilet paper roll.
Sources of Error
However, this equation is not going to be a completely accurate method. There are still many sources of error like how much the toilet paper gets compressed or maybe even the gaps of air in between each wrap. Then there is the important factor which is the fact that the toilet paper sheets are wrapped in a spiral rather than concentric circles. But overall, this equation yields a good approximation.
