Pizza Economics aka "Proof-of-Pizza" Equation

Cowabunga dudes!

Let's say you have a bunch of people coming to an event and you were in charge of ordering food. The fastest and simplest thing that comes to mind is pizza. Now the next step is determining how many boxes of pizza to order and how much it will cost. Determining how many people are coming also helps because that is one of the variables to use when estimating the costs to feed them. This is based on real life events from last minute situations when ordering food to feed guests. Often times it is something that is quick to deliver or pick up like pizza. It is also a universal favorite, though it is about ordering the right kind (plain cheese for non-meat eaters and supreme for those who love their protein) .

Let's say we have a gathering of 24 people. To make sure everyone gets their fill, let's assume 2 slices per person. No preference for flavor. Thus we can say:

N = No. of people
X = No. of slices for each person
T = Total No. of slices

N = 24
X = 2
T = ?

We will need to have a total of:

T = N x X = 24 x 2 = 48 slices

According to the pizza menu, the ff. are the prices with the size of the pizza for premium flavor:

X-Large 16” = $17.99
Large 14” = $15.99
Medium 12” = $13.99
Small 10” = $11.99

The estimated number of slices per box depends on the size. So the following shows how many slices are in each box size with the total price required to feed N people. I already computed that based on:

B = No. of boxes
S = Slices per box
B = T / S

I have also added the total area or size dimension of the pizza (Multiplied the area of each box by total number of boxes) based on its given diameter. We must take the radius of each size to determine that.

A = Area of pizza pie
A = πr^2
A = 3.1416(radius)^2

S
4 slices/box
12 boxes x $11.99 = $143.88
942.48 sq. inches

M
8 slices/box
6 boxes x $13.99 = $83.94
678.54 sq. inches

L
12 slices/box
4 boxes x $15.99 = $63.99
615.72 sq. inches

XL
16 slices/box
3 boxes x $17.99 = $53.97
603 sq. inches

From the data provided, it is more economical to buy 3 XL boxes of pizza at $53.97 which will feed 24 people with 48 slices at 2 slices/person. However, ordering 12 S might be more expensive, but in terms of actual quantity it contains more pizza at 942.48 square inches (78.54 sq. inches per box multiplied by 12 boxes). There is always a tradeoff with price and quantity. If budget is an issue, then ordering the largest size to divide up for the guests is the best option. If the guests are really hungry and demand the most pizza they can get, ordering the 12 small boxes to be shared by 2 people each, may be the best choice.

To prove this we have to calculate the circumference of the pizza pie. So let's compare the S 10" to the XL 16".

C = 2πr

The S 10" pizza has a C = 31.4159" which when divided by 4 slices is 7.85" per slice (C/S). Compare to XL 16" pizza which has a C = 50.26" divided by 16 slices is 3.14" per slice. Now the fun part, we can measure the area per slice using its Length(1/2 the diameter per slice) and Width(C/S).

S = 5" x 7.85" = 39.25 sq. inches
XL = 8" x 3.14" = 25.13 sq. inches

In the end everybody eats some good pizza.