## Pizza Math

September 2nd, 2014 by Potato*A reader requested this a long time ago, sorry for taking so long Ben!
*

The age-old question: is the medium the better deal, or the large? The medium may be cheaper per slice, but each slice on the large is bigger…

The math to figure this out is not hugely complicated, but it’s just a bit more than you might be able to do in your head or with a smartphone while you’re hungry and staring at a menu board. What we’re interested in is the area of pizza that you get per dollar. The area of a circle is simply pi * r^{2}. Pizzas are sized by their diameter (double the radius). However, there are no points for crust (“pizza bones”), so we’ll subtract 1″ from each diameter (for a typical 0.5″ of crust on each side of the line through the circle) when computing the area factor. Because we’re really just interested in the relative value we don’t necessarily need to do the division by two or multiplication by pi — the pizza value will scale with the square of the adjusted diameter — unless we’re comparing to a square pizza. While some pizza places use their own wacky sizes, or have irregular hand-shaped crusts, most places have settled on standard sizes. I’ve listed the rounder area factors and actual edible areas below:

Small (nominally 10″): Usable diameter of 9″, area factor is 81 (edible area of 63.6 sq. in.).

Medium (nominally 12″): area factor is 121 ( 95 sq. in.).

Large (nominally 14″): area factor is 169 (132 sq. in.).

Extra Large (nominally 18″): area factor is 289 (227 sq. in.).

(note that Pizza Pizza and some other stores have 16″ extra larges)

Square pizzas: most often encountered with party sized pizzas. In this case to make a true comparison you would need the circular pizzas area in square inches. For a 15×21″ (nominal) party pizza, there are 280 sq. in. of edible pizza. Converting into “area factor” above, that would be 356.

To put this into practice then requires a division step with the price. You can divide the price by the area factor to get a price per unit area — then lower is better. However, because pizzas are often priced near $10 or $20, the inverse may be more convenient to work with — pizza units per dollar — in which case the higher the number the better value. For example, if a large is on for $10, the pizza per dollar is 169/$10 = 16.9. If the medium is $8, that’s 121/8 = 15.1; if the party size is $20 that would come to 356/20 = 17.8. In that case the bigger you go, the better your value.

For your convenience, I made a reference card for your wallet. (Be sure to select “actual size” when printing)

I’ll note that dollar per unit pizza should be the preferred unit/method if you want to look at how the value difference scales across pie sizes rather than just which is larger — analogous to the L/100 km measurement system vs MPG issues.