Pizza Pi: How Pi Can Help You Get The Best Deal On Pizza

Whether or not you're going to be celebrating Pi Day on March 14 (i.e. 3/14), you can use the famous transcendental constant to help you get the best bang for your buck at the pizzeria. If you're picking up some pizza to share with friends, you probably feel like two 12-inch pizzas would be a better deal than a single 18-inch pizza, but you would be wrong. To find out why, you need to learn to use pi and the formula for the area of a circle to your advantage.

The formula for the area of a circle is one of the most well-known equations that makes use of pi:

\(A = πr^2\)

Where A stands for the area and r is the radius of the circle. This is the key to turning those pizza sizes into the actual amount of pizza you get, in terms of the area of a circle. The area is proportional to the square of the radius. So if circle A has twice the radius of circle B, it will occupy four times as big an area.

The downside to this formula when we're thinking about pizza (which, I'll be honest, I always am) is that pizza sizes are expressed in diameter (d). This is just twice as big as the radius, so you can either convert a pizza diameter into a radius and use the formula above, or change it to suit pizza:

\(\begin{aligned}
A&=\pi r^2 \
&=\pi \bigg(\frac{d}{2}\bigg)^2 \
&=\frac{\pi d^2}{4}
\end{aligned}\)

Using either of the formulae above and comparing areas, you can work out whether it's better to get two 12 inch pizzas or one 18 inch pizza if the price works out the same. Give this a try before reading on if you want to work it out for yourself.

For one 12-inch pizza, the second formula gives:

\(\begin{aligned}
A&=\frac{\pi d^2}{4} \
&= \frac{\pi × (12 \; \text{inch})^2}{4} \
&= \frac{3.14159 × 144 \;\text{inch}^2}{4} \
&=113.1 \;\text{inch}^2
\end{aligned}\)

Since you're getting two, you'd end up with 113.1 inch² × 2 = 226.2 inch² of pizza.

Using the first formula, an 18 inch diameter pizza has a radius of r =18 inch / 2 = 9 inch. So:

\(\begin{aligned}
A &= π × (9 \;\text{inch})^2 \
&= 3.14159 × 81 \;\text{inch}^2 \
&=254.5 \;\text{inch}^2
\end{aligned}\)

This area is bigger than that of two 12-inch pizzas, so you get more pizza with the single 18-inch. If they're the same price, you should definitely get the 18-inch.

If you have to compare different size pizzas with different prices, a simple area comparison like in the previous section won't give you enough information to make your choice. You can compare them in a rough way by just comparing the areas and corresponding prices, but the easiest method is just calculating the price per square inch.

Imagine that a 10-inch diameter (5-inch radius) pizza costs $6.99. The area of the pizza is:

\(\begin{aligned}
A &= π × (5 \;\text{inch})^2 \
&=78.54 \;\text{inch}^2
\end{aligned}\)

Price per square inch is given by:

\(\text{Price}/\text{inch}^2 = \frac{\text{Total cost}}{A}\)

So for the 10-inch:

\(\begin{aligned}
\text{Price}/\text{inch}^2 &= \frac{\$6.99}{78.54 \;\text{inch}^2} \
&=\$0.089/\text{inch}^2
\end{aligned}\)

Using this approach, you can compare value for money for various pizza sizes and prices. At the same pizzeria as the $6.99 for 10-inch pizza calculated as $0.089 / inch², you can also get a 13-inch for $9.99, a 16-inch for $12.99, an 18-inch for $14.99, a 24-inch for $22.99, a 28-inch for $28.99 or a huge 36-inch for $44.99. Which is the best value for money?

The best way to work this out is to make a table like this:

\(\def\arraystretch{1.5}
\begin{array}{c:c:c:c}
\text{Size/inches} & \text{Price/\$} & \text{Total Area/sq. inch} & \text{Cost per sq. inch} \ \hline
10 & 6.99 & 78.54 & \$0.089 \
\hdashline
13 & 9.99 & & \
\hdashline
16 & 12.99 & & \
\hdashline
18 & 14.99 & & \
\hdashline
24 & 22.99 & & \
\hdashline
28 & 28.99 & & \
\hdashline
36 & 44.99 & &
\end{array}\)

Use the method in the previous section to work out which pizza gives the best value for money, and you can see how much pizza you'll end up with using the total area column, too.

Here are the results:

\(\def\arraystretch{1.5}
\begin{array}{c:c:c:c}
\text{Size/inches} & \text{Price/\$} & \text{Total Area/sq. inch} & \text{Cost per sq. inch} \ \hline
10 & 6.99 & 78.54 & \$0.089 \
\hdashline
13 & 9.99 & 132.73 & \$0.075\
\hdashline
16 & 12.99 & 201.06 &\$0.065 \
\hdashline
18 & 14.99 &254.47 & \$0.059\
\hdashline
24 & 22.99 &452.39 & \$0.051 \
\hdashline
28 & 28.99 & 615.75& \$0.047 \
\hdashline
36 & 44.99 & 1017.88& \$0.044
\end{array}\)

So the bigger the pizza, the better the deal. The biggest pizza is less than half of the cost of a 10-inch per square inch, and you get almost 13 times as much pizza for around 6.4 times the cost.

Now for the real challenge: working out how much pizza you can eat without putting yourself into a food coma.