First of all, let’s get one thing out of the way: Archimedes never used the Greek letter “pi” when he used it to calculate the area and circumference of a circle. No, never; so just by reading this blurb you've learned something new!
The point of this activity is threefold: the first is to show that as you double the diameter of a circle, the area of that circle would quadruple. That’s a very important concept, because many of your students have only experienced relationships where if one variable is doubled, so is the second (for example, if you double the amount of ingredients you use in a pie recipe, you’ll get double the amount of pie, although you definitely should not double the temperature at which you bake it....)
The second goal is to show that pizza prices should be calculated on a “price per square inch” basis, and not simply on the dimater alone. That is, it may make sense for a 4” pizza to be priced at $4, a 6” pizza at $6 and an 8” pizza at $8, because it means that it is $1 for each inch of diameter. But that kind of pricing does not make sense, because the diameter of the pizza does not have a linear relationship with the amount of pizza you actually get.
The third goal of this activity is to show that it is impossible to turn a circular pizza into a square pizza. This is the old problem “how can we square the circle?” which has been proven impossible to solve. However, you can get fairly close. To lessen the frustration that students may have with this section of the investigation, you may want to have them use a calculator (as well as with the “price per square inch“ activity as well..)
This is a really cool activity and it blows kids minds every time when they find out that by doubling the diameter of a pizza (which is how pizzas are measured), they are actually get four times as much pizza to eat. When you consider that cold pizza is one of life’s great pleasures, buying the largest pizza possible and then taking home the rest for the next day is always the way to go.