Geometry

Posted Friday, April 16 2010 by jonathan

I was just challenged to justify ordering a 14" deep dish pizza from our local pizzeria rather than the smaller, more reasonable 12" version. It all comes down to volume in the end. I'm a fan of volume (more a fan of leverage, but volume is important too).

So I started by establishing that we could agree that the pizza was sufficiently circular and the depth of the toppings was, more or less, constant across the entire pie. In exchange for these kindnesses my challenger eliminated pencil, paper, and computing device from my available tools.

The volume of a cylinder can be calculated as a function of it's radius and height. Pi * R^2 * H where R is the radius of the pizza and H the depth of toppings (height, in other words). Now, since multiplying by H is a linear function and we're comparing two pies with differing radii but equal depth we can eliminate the term entirely to simplify the in-my-head-math and just compare surface area. We can do the same for Pi, transforming our calculation of surface area into a simple "size factor".

So the size factor of a 12" pie is 12 squared, or 144. The size factor of a 14" pie is 196. The difference is 52, an increase of somewhere around 30%. If you want to know the actual difference in surface area, multiply by Pi to get approximately 161 square inches. To calculate the actual volume of food gained multiply by the average depth, .75, to get 120 cubic inches of food.

"But wait! The outside two inches of the pizza is just crust! That's not viable volume, that's like selling a thousand hogs heads of soda at a loss."

Ok, totally fair. Luckily both 12 and 14 inch pizzas appear to have an equal crust-radius, approximately 2 inches. So we can subtract 2 inches from R and try again.

12" pie = (12 - 2) ^2 = 100
14" pie = (14 - 2) ^2 = 144

A difference of 44 units, which is an increase of 44%. Even more reason to order the larger size!

For those who can't leave these things half-done, that's a net gain of 44 * Pi * .75 = 105 cubic inches of the good stuff.

To generalize, reduce, and recap, we have established that the difference in topping-value between two sizes of deep dish pizza, A and B, may be calculated as a function of the radii Ra and Rb of said pizzas with this formula:

Vd = ((Rb-2) ^ 2 - (Ra-2) ^ 2) / ( (Ra-2) ^ 2)

If Vd exceeds the relative increase in price between pizzas A and B [Pd = (Pb - Pa) / Pa], it's a good deal. (But like EBITDA we have not accounted for painful overindulgence in CapEx.)

And they asked what good geometry was in high school.

Your Thoughts?