Suppose you are at a pizza restaurant with your friends. You all agree you want to buy pizza to maximize your pizza-per-dollar. There’s an easy way to make comparisons between pizza sizes and figure out what the best deal is.

First, note that this won’t be possible in places that just do a generic “Small, Medium, Large” set-up (like your Dominoes or Papa John’s.) You need the actual size of the pizza in inches.

Here’s an example of one-topping prices from a local pizza place of mine:

10″ : $14.49 12″ :$18.29

14″ : \$21.59

At a glance it might be hard to figure out what your best deal is. The naive mathematical way is to remember that the area of the pizza is $\pi r^2$. Our 10″ pizza has a radius of 5″, so its area is $\pi\cdot 5^2 = 25\pi \approx 78.5 \text{ in}^2.$ Similarly, the area of the 12″ is $113.1\text{ in}^2$, and the 14″ is $153.9\text{ in}^2.$

Then, we divide by the price to determine the amount of pizza we get per dollar spent (ppd). In this case, we have

10″ : 5.4 ppd

12″ : 6.1 ppd

14″ : 7.1 ppd

A larger number is better in this metric (assuming you want more pizza for your money), so the 14″ is the better deal. This is pretty typical across the food industry, that as you increase in size you get a better deal. There are many reasons for this that are not our concern. We’re assuming you’re already ordering a certain amount of pizza, and to decide if you should, for example, get two 10″ pizzas with different toppings, or just split a 14″ pizza. Looking at the raw numbers, a 10″ pizza is about half the size of a 14″ pizza (78.5 vs 153.9 square inches.) So it would be better to go through the trouble of splitting a 14″ pizza.

The above reasoning totally works, but is impractical. Why bother multiplying by that pesky $\pi$? Notice that if I directly compare the size of two pizzas, say a pizza with radius $r$ and another with radius $R$, I get the following:

$\dfrac{\text{area 1}}{\text{area 2}} = \dfrac{\pi r^2}{\pi R^2} = \dfrac{r^2}{R^2}$

That $\pi$ just disappears! Since it is just a scale factor, we can ignore it for the comparisons we are making. Now we can just deal in “pizza area units”, where we only look at the radius squared. Hence, a 10″ pizza is $5^2 = 25$ pizza units, and a 14″ pizza is $7^2 = 49$ pizza units. Not only is this faster, it actually makes it much more obvious that a 14″ pizza is nearly twice as large as a 10″ pizza.

We can do the same calculations. A 10″ pizza from my local pizza place now has a value of $25/14.49 = 1.7$. Similarly, the 12″ has a value of 2.0, and the 14″ has a value of 2.3. Again we get a better deal from the 14″ pizza, with the intermediate values scaling in the same way (increasing by about 0.3 each time, instead of about 1.0 each time above. But this makes since, as $0.3 \cdot \pi \approx 1.$) With some approximations it’s possible to get close estimates in your head.

So, next time you’re in the mood for pizza, this is a simple approach to get your best deal. It’s also a good way to figure out how much pizza you may want. Do two people want a 12″ pizza? Each 12″ pizza has 36 pizza units. Meanwhile, a 16″ pizza is 64 pizza units, and an 18″ pizza is 81 pizza units. So, it would probably end up being a better deal to split the 18″ pizza, and you’d get some extra pizza out of the deal.

As a side note, this also works for cakes, pies, and any other circular treats you may enjoy.