## A math koan

One sheet of toilet paper is `L` long. The inner circumference of the roll is `C1` and the outer circumference is `C2`. Each sheet is `T` thick.

How many sheets are there on a fully-loaded roll? Please post here if you have an answer.

And no cheating by peeking at the package. ;)

Technorati tags: Math, Toilet paper. (Did you expect to ever see those two tags together in one post?)

June 28th, 2006 at 17:24:00

(c2^2 – c1^2)pi/LT

June 28th, 2006 at 17:30:00

OMG, I love math problems! This one’s a cinch.

We’re going to do it by volume. Basically, we’re going to compute the total volume of toilet paper in one roll, and then divide it by the total volume in one sheet of toilet paper. This is valid since when you use toilet paper, there is no paper wasted. Well, strictly speaking, you’re wasting it by using it, but you know what I mean. (From now on, I’m going to call a “sheet” a “square” of toilet paper instead. “Sheet” seems to imply the whole roll — i.e.: in 2-ply toilet paper you have 2 sheets, whereas with crappier toilet paper you have only 1 sheet. But I digress.)

So, the total volume of one square of toilet paper is simple. Length times thickness times width. Crap! You didn’t provide a variable for width of the toilet paper. But we’ll see that that doesn’t matter in the long run. So let’s define a temporary variable called W. The volume of a single square of toilet paper is therefore L*T*W.

Now for the total volume of the roll. The outer radius R_2 = C_2/(2*pi), since C_2 = 2*pi*R_2. (I’m using C_1 and C_2 in place of C1 and C2 so that you don’t accidentally confuse with multiplication.) That means the area of a cross section of the roll including the hole is C_2^2/(4*pi), since the area would be pi*R_2^2. Similarly, the area of a cross section of the hole is C_1^2/(4*pi).

So the cross sectional area of just the toilet paper is going to be the total cross sectional area minus the cross sectional area of the whole, which is (C_2^2 – C_1^2)/(4*pi).

Now for the total volume of toilet paper: just multiply by the width of the roll. This is just the same as the width of one square of toilet paper, which means we’re just working with the same variable we defined earlier called W. So the total volume of toilet paper is (C_2^2 – C_1^2)*W/(4*pi).

And finally, we divide the total volume by the volume of a single square of toilet paper:

(C_2^2 – C_1^2)*W/(4*pi) / L*T*W

Well, what do you know! The W’s cancel out! This intuitively makes sense because if you were to take a roll of toilet paper and stretch it so it has a greater width, you’d still have the same number of squares in the roll. So the number of squares of toilet paper in a roll is given by:

(C_2^2 – C_1^2)/(4*pi*L*T)

where C_2 is the outer circumference, C_1 is the inner circumference, T is the thickness of a square of toilet paper, and L is the length of a square.

Yay! (Unfortunately, I don’t have any toilet paper on which to test this out.)

June 28th, 2006 at 17:33:00

Oops, I had the radius labeled as the circumfence in my diagram. Rather, it would be (c2^2 = c1^2)/4piLT

June 28th, 2006 at 17:37:00

Oh, incidentally, in case you actually WERE asking a way to figure out the number of ply in a roll rather than the number of squares, the answer would be identical. Since L would mean the length of the sheet (as you said), then L*T*W would still be the volume of a single sheet (ply) of toilet-paper, in which case you’d use the exact same method. Of course, in this case, your variables would be kind of weird since you’d have to unroll the whole roll to get at the length of a sheet (ply) of toilet paper.

If you didn’t want to unroll the roll, and you WERE asking about how many ply in a roll of toilet paper, you’d need an additional variable: how many squares are in a single sheet. In this case, L would refer to the length of a square rather than the length of an entire sheet.

But I’m pretty sure you meant sheet as in square, not sheet as in ply. :)

June 28th, 2006 at 18:30:00

Simone: Yes, your definition of “square” is what I meant. I didn’t say “square” since the width of each sheet is irrelevant; you could very easily have 8×1″ sheets. :)

June 28th, 2006 at 18:31:00

And since you both seem to have independently agreed upon the same answer (I approved all four comments at the same time), I’m going to just assume that that’s right and not unroll this roll of toilet paper. ;)

June 28th, 2006 at 20:01:00

So was this just a random curiosity, or was there a reason you wanted to know? :P

(And two other notes: perhaps “segment” is a better description for my “square” and your “sheet”.

Second, this method really isn’t that reliable because it’s really hard to accurately measure the thickness of a segment. I tried doing it with a new roll of toilet paper, and I got 379 segments vs. 425 as indicated on the wrapper. And it’s also complicated by the fact that a toilet paper roll — and perhaps even a segment — could have its measurements thrown off by air in between wrap-arounds and perhaps in between plys. The former problem is probably far worse, though.)

June 28th, 2006 at 20:12:00

Simone: Random curiosity. :)

“I got 379 segments vs. 425 as indicated on the wrapper.” Maybe you should go unroll that roll, and write a letter to P&G if it turns out that you were right. :)

June 28th, 2006 at 20:18:00

“Maybe you should go unroll that roll, and write a letter to P&G if it turns out that you were right. :)”

Psh! I’m not THAT bored. (… says the person who just worked through the calculations and tried to verify it. ;) )