RL Maths: The Great Pizza Ripoff

I went to a well-known pizza restaurant chain for dinner last night. I won't tell you which one, but I'll refer to it as 'Pizza Shed' and you can read from that what you will. When we were ordering, my girlfriend and I briefly wondered whether we should get a regular pizza each, or a large pizza with 'half-and-half' toppings. We went for the latter, not really thinking too much about it (we wanted a stuffed crust anyway).

However, when the bill came we were surprised to note that our half-and-half large pizza had been charged as two regular ones. This got us thinking: is one large pizza really worth the same as two regular ones? The good thing is that we don't even need to know the price to work this out. We do need to know the sizes, though: a regular pizza of the type we ordered is 11" across, and a large is 14".

At first glance, you might say "the large pizza isn't twice the size of the regular one, so no, it's not worth it!" but be careful: you're making a mistake that loads of people make when considering the mathematics of pizza ordering*.

Consider the hastily produced graphic to the left**. It shows two small pizzas, side by side, plonked on top of a large pizza. The important bit is that the two small pizzas together have the same diameter as the large pizza, but when plonked on top there's still a large expanse of pizza that isn't hidden. This means that twice the diameter means much more than twice the pizza.

So we've got to be a bit more cleverer about this. When talking about how much pizza you get, it makes more sense to talk about the area of a pizza than its diameter (which is the standard quoted measurement for pizzas worldwide). If you've got a maths GCSE (or O-Level) you'll have at least a dim recollection that the area of a circle and its radius (which is half the diameter) are linked. Specifically, the formula is:

A = π x r²

Where A stands for 'area' and r stands for 'radius'. And π, far from being something scary, is just a symbol that means "a bit more than 3"***.

So, for our pizzas, we can find the regular one's area by working out π x 5.5², and finding the answer to π x 7² will give us the area of a large pizza.

A good idea in maths is to estimate things before working them out exactly. This gives you a feeling for what kind of number you're expecting the actual answer to be, so you can tell if you've messed something up (or hit the wrong button on the calculator if you've got fat fingers like me). In maths 'estimating' is not just guessing, but working out a simpler version of the problem. I did this, last night, in a minute or so whilst driving home (that's how easy it is****).

So the regular pizza has an estimated area of "about 3 multiplied by about 30*****", which is about 90 square inches, and the large pizza has an estimated area of "about 3 multiplied by about 50******", which is about 150 square inches. It looks like two regular pizzas gives you more total pizza than one large one!

What's the solution, exactly?

• π x 5.5²  = 95.03 square inches (to 2dp).
• π x 7² = 153.94 square inches (to 2dp).

My estimates were both slightly low, but not by a lot. The same result stands: two regular pizzas would have given us more pizza than one large one, yet they charged us the same amount!

The scoundrels!

* O.k, the number of people who actually stop to consider the mathematics of pizza ordering might be quite small.
** I am available for freelance graphic design jobs for a modest fee.
*** π is, of course, a specific number (3.14159... etc), but to write it down accurately would take more time and energy than is available to us, even if we enlist our children's children's children, so we're lazy and use a symbol instead.
**** If you'd like some tips for being more accurate with your mental maths for not much extra effort, you could do worse than check out my pal Colin's 'Mathematical Ninja' series.
***** 5² is 25, and 6² is 36, so 5.5² is about half way between, which is about 30.
****** 7² is 49, which is about 50.