Math in… Toilet Paper?

When you use a roll of paper, like toilet paper, you might notice that you use it up faster towards the end of the roll.

The cardboard tube in the middle of a roll has a diameter of about 1.5 inches. Depending on what kind of toilet paper you buy, your roll’s overall diameter will usually be between 4 and 6 inches.

Using the side of the roll, we can use math to determine how much toilet paper is left!

If we measure the diameter of the roll, we can find the radius and draw a diagram.

In this diagram, the radius of the whole roll is half of 4.5 inches, and the radius of the remaining orange roll is half of 3 inches.

Then, we can use our formula for the area of a circle:

To find the area of the green and orange parts together, we can plug our values into the formula, and subtract the hole in the middle.

We can also use our measurements to find the areas of the green and orange parts separately!

The green and orange parts have the same thicknesses, which makes it look like they would have the same amount of toilet paper. But actually, the orange part only has

of the toilet paper left!

What would happen if the cardboard tube was bigger? Smaller?