Value Example: Pizza

TLDR

Larger pizza sizes are often way better value than smaller pizza sizes.

You can use this equation to see how much better the value is in terms of square inches per dollar.

Let Pizza 1 be the larger one with price p1p_1 and diameter d1d_1. Let Pizza 2 be the smaller one with price p2p_2 and diameter d2d_2.

Value of Pizza 1 Compared to Pizza 2=p2∗d12p1∗d22 \text{Value of Pizza 1 Compared to Pizza 2} = \frac{p_2*d_1^2}{p_1*d_2^2}

Read on for all the details!

Gioia’s Pizza in SF

Sorry, this is about value and not expected value. There’s an assumption that at a pizza restaurant ordering by the full pizza is a better deal than ordering by the slice and also that ordering larger pizzas is a better deal than ordering smaller ones.

Last week I ate my first San Francisco pizza at Gioia’s. I of course got a margherita pizza (I don’t eat meat on pizza and don’t like vegetables on pizza cause then they remind me of omelettes (I also don’t like vegetables in omelettes)). [Note: I now found out that they actually have a plain cheese pizza that is even more plain and chepaer!]

The menu was like this:

Pizza Type Price Notes
14” Pizza $24 6 slices per pizza
18” Pizza $34 8 slices per pizza
18” Pizza Slice $4.50 Slice from 18”

General Pizza and Circle Computatations

Pizzas are approximately circular. The equation to calculate the area of a circle is below. Note that pizzas are shown with their diameter, but the equation uses radius, so we have to divide the diameters in 2 before using them.

Area=π∗r2 \text{Area} = \pi * r^2

Area of Pizzas

Let’s get the area of each pizza so they are directly comparable.

14" Pizza Area=π∗72=153.94 in.2 \text{14" Pizza Area} = \pi * 7^2 = 153.94 \text{ in.}^2

18" Pizza Area=π∗92=254.47 in.2 \text{18" Pizza Area} = \pi * 9^2 = 254.47 \text{ in.}^2

There are 8 slices in the 18” pizza, so we can compute the area per slice: 18" Pizza Slice Area=254.47/8=31.81 in.2 \text{18" Pizza Slice Area} = 254.47/8 = 31.81 \text{ in.}^2

Let’s review:

Pizza Type Price Square Inches Notes
14” Pizza $24 153.94 6 slices per pizza
18” Pizza $34 254.47 8 slices per pizza
18” Slice $4.50 31.81 Slice from 18”

Price per Area of Pizzas

Now let’s run the numbers on the square inches per dollar.

14" Pizza Sq In. per Dollar=153.94/$24=6.41 \text{14" Pizza Sq In. per Dollar} = 153.94/\$24 = 6.41

18" Pizza Sq In. per Dollar=254.47/$34=7.48 \text{18" Pizza Sq In. per Dollar} = 254.47/\$34 = 7.48

18" Slice Sq In. per Dollar=31.81/$4.50=7.07 \text{18" Slice Sq In. per Dollar} = 31.81/\$4.50 = 7.07

Pizza Type Price Square Inches Square Inch per Dollar Notes
14” Pizza $24 153.94 6.41 6 slices per pizza
18” Pizza $34 254.47 7.48 8 slices per pizza
18” Slice $4.50 31.81 7.07 Slice from 18”

Direct Comparisons

From the above chart we see that the 18” pizza for $34 is the best deal because it has a value of 7.48 sq in. per dollar.

So if you want 8 large slices, that’s the way to go.

The 14” pizza is the most expensive option at $24 for the pizza, getting only 6.41 sq in. per dollar.

Therefore if you want less than 8 large slices, it’s probably best to just order by the slice at 7.07 sq in. per dollar.

How many slices from the 18” pizza would it take to match the size of the 14”? The 14” is 153.94 sq in. and the 18” slices are 31.81 sq in.

How many slices?=153.94/31.81=4.84 \text{How many slices?} = 153.94/31.81 = 4.84

How much would it cost for 4.84 slices?

Cost of 4.84 slices=$4.50∗4.84=$21.78 \text{Cost of 4.84 slices} = \$4.50*4.84 = \$21.78

This means we can get the same amount of pizza for $21.78, except we can’t actually order 4.84 slices. So how about 5?

Cost of 5 slices=$4.50∗5=$22.00 \text{Cost of 5 slices} = \$4.50*5 = \$22.00

The small pizza is a terrible deal! You can get 5 slices from the 18” pizza, which is slightly more pizza area (31.81∗5−153.94=5.1131.81*5-153.94 = 5.11 bonus sq in.) and is $2 cheaper!

And finally, if you spent the $24 you would have on the small pizza on slices, how many could you get?

Slices for $24?=$24/$4.5=5.33 \text{Slices for \$24?} = \$24/\$4.5 = 5.33

Therefore you would have 5.33∗31.81−153.94=15.615.33*31.81 - 153.94 = 15.61 bonus sq in., or about half a slice bonus for the same price.

Other Restaurants

How to not make this mistake yourself? The above situation seems fairly rare.

The more common scenario is that larger pizzas are actually way cheaper than smaller ones per sq in and so the error is to ever get a small pizza.

Here are some prices for Domino’s (most well known) and Lou Malnati’s (finest).

Pizza Type Price Square Inches Square Inch per Dollar
Domino’s 10” $12.99 78.54 6.05
Domino’s 12” $15.99 113.10 7.07
Domino’s 14” $18.99 153.94 8.11
Pizza Type Price Square Inches Square Inch per Dollar
Lou Malnati’s 6” $10.59 28.27 2.67
Lou Malnati’s 9” $15.99 63.62 3.98
Lou Malnati’s 12” $21.49 113.10 5.26
Lou Malnati’s 14” $26.79 153.94 5.75

As you can see, the square inch per dollar values tend to go up significantly as pizzas get larger. You might think that getting two 6” pizzas instead of 12” makes sense, but the area equation is based on the square of the radius, and 323^2 is much smaller than 626^2 (it’s 1/41/4!).

At Lou Malnati’s if you got two 6” pizzas, you’d be paying $21.18 for 56.54 sq in. of pizza, compared to a 12” for $21.49 that gives you 113.10 sq in. of pizza. The bigger one is exactly double the size for just about the same price!

Whole Foods

At Whole Foods, a whole 18” cheese pizza is $12, which is 6 slices. By the slice it’s $8 for 2 or $4.29 each. In this case it’s quite clear that the full pizza is a better deal. It would cost $24 to buy 6 slices if buying in pairs compared to $12 for the same amount if buying a full pizza!

General Equations

The general equation for sq in. per dollar is:

Sq In. per Dollar=AreaPrice=π∗r2Price \begin{equation} \begin{split} \text{Sq In. per Dollar} &= \frac{\text{Area}}{{\text{Price}}} \\ \\ &= \frac{\pi*r^2}{\text{Price}} \end{split} \end{equation}

If you want to compare two pizzas and you know the sizes and prices, you can use a shortcut and take the ratios.

Let’s formalize and clarify this for general pizza ii with diameter dd, radius d/2=rd/2 = r, and price pp

Sq In. per Dollar=π∗(di2)2pi \text{Sq In. per Dollar} = \frac{\pi*(\frac{d_i}{2})^2}{p_i}

To compare two, we can take the ratios:

Sq In. per Dollar=π∗(d12)2p1π∗(d22)2p2=π∗p2∗(d12)2π∗p1∗(d22)2=p2∗d12p1∗d22 \begin{equation} \begin{split} \text{Sq In. per Dollar} &= \frac{\frac{\pi*(\frac{d_1}{2})^2}{p_1}}{\frac{\pi*(\frac{d_2}{2})^2}{p_2}} \\ &= \frac{\pi*p_2*(\frac{d_1}{2})^2}{\pi*p_1*(\frac{d_2}{2})^2} \\ &= \frac{p_2*d_1^2}{p_1*d_2^2} \end{split} \end{equation}

Note that π\pi and the denominator of d/2=rd/2 = r drop out, so we only need to use p1p_1, p2p_2, d1d_1, and d2d_2.

The Comparison Equation

This is the important equation that you should use in real life pizza value scenarios!

Let Pizza 1 be the larger one for consistency.

Value of Pizza 1 Compared to Pizza 2=p2∗d12p1∗d22 \text{Value of Pizza 1 Compared to Pizza 2} = \frac{p_2*d_1^2}{p_1*d_2^2}

Here’s an example using Lou Malnati’s 12” and 6” pizzas from above:

Pizza Diameter Price
Pizza 1 12” $21.49
Pizza 2 6” $10.59

Now we can use our equation:

Value of Pizza 1 Compared to Pizza 2=p2∗d12p1∗d22=$10.59∗122$21.49∗62=1.97 \begin{equation} \begin{split} \text{Value of Pizza 1 Compared to Pizza 2} &= \frac{p_2*d_1^2}{p_1*d_2^2} \\ &= \frac{\$10.59*12^2}{\$21.49*6^2} \\ &= 1.97 \end{split} \end{equation}

This shows us that Pizza 1 is about 1.97x better value than Pizza 2 (in terms of square inches per dollar)!