Let's say a pizzeria charges for their pies proportional to the size of the pie. If a 10" pizza costs $10, then what should a 12" pizza cost?

It’s tempting to think that a 12-inch pizza, being 20% wider than a 10-inch one, should cost 20% more, or $12. However, the amount of pizza you get is determined by its area, not its diameter. Our brains often get tricked by linear measurements, but the ingredients—dough, sauce, and cheese—cover a two-dimensional surface. This means we have to compare the areas of the two pies to find the fair price.

The area of a circle is found using the formula πr², where 'r' is the radius (half the diameter). The 10-inch pizza has a 5-inch radius, and the 12-inch pizza has a 6-inch radius. The key is that the area scales with the square of the radius. To find how much larger the 12-inch pizza is, we can compare the squares of their radii: 6² divided by 5², which is 36 divided by 25. This gives us a ratio of 1.44. The larger pizza has 1.44 times the area, meaning it's 44% bigger, not 20%.

This principle shows up in many places beyond the pizzeria. It’s why a 40-inch television has four times the screen area of a 20-inch one, not double. When an object's linear dimensions are scaled up, its area scales up by the square of that factor. So, if the pizzeria is pricing its pies fairly based on the amount of food you receive, the 12-inch pizza should cost 1.44 times the price of the 10-inch one, leading to the correct price of $