A sleepy high school class typically gives an incorrect answer to the following question:

If a 10-inch pizza sells for $10, how much should you pay for a 20-inch pizza, assuming that price only depends on mass?

Their typical response is $20. But as you know, the correct answer is $40 because if price is proportional to mass then price depends on area and is therefore proportional to the square of the pizza’s diameter.

If you want to recreate the look on a patient facing a dentist’s drill, simply ask students to sit through the simplest of mathematical derivations. But since our task as teachers is not to entertain but to get students to think while we communicate as effectively as we can, we should love to reveal the origin of formulas.

Mass is the product of density and volume. A large pizza and a small one have the same density. So if the price is proportional to mass, then price is proportional to the volume of pizza.

A pizza is approximately cylindrical in shape, so its volume = π r^{2 }h, where r = radius and h = height. But pizza-height remains constant; only its diameter(D) changes. Express the radius as half the diameter, and the above formula simplifies to the following:

Although pizzerias would go out of business if they priced their product according to the logic of a sleepy class, they actually sell larger pizzas for less than the mathematical price. Psychologically a to-the-square-scaled price would discourage consumers. By discounting the price for larger pizzas, they encourage people to buy bigger. Although the profit margin per square inch is smaller, the margin per pizza sold is higher. (My spreadsheet reveals that one particular merchandiser, however, made a mistake in pricing the 28″ pizza)

Elsewhere in the economy, such a pricing strategy encourages overconsumption, which has ecological and health ramifications. But when pizza is shared among family and friends, potentially an everyone-wins-situation could arise.

For example, a typical 10″ pizza provides about 821 kcal, a recommended intake for one meal for an active adult. It’s equivalent to about 10.5 kcal per square inch. Compared to a single customer, a group of four would need four times the pizza-area; ideally they should buy one with the square root of 4 or twice the diameter to get the same caloric needs. Unfortunately, the above merchant does not offer a 20″ pizza. The 18″ one has only (18/20)^{2 } = 81% of the required calories while the 24″ one offers a 44% excess for a group of four.

Another factor to consider is that not all pizza toppings have the same carbon dioxide equivalent per kilocalorie eaten. Toppings with only oil and vegetables create a lower ecological footprint than those with cheese and meat, especially if the latter includes beef. As “*Shrink That Footprint*” points out, the graphics don’t reveal that out-of-season fruit and other meats can also have relatively high carbon dioxide emissions. Finally, the way animals are raised is another factor with implications for carbon intensity of both meat and dairy consumption.

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