Felt a little stupid last night when my pizza was delivered and it was way too small. But of course pizza places don't tell you the areas of the pizzas, which is precisely the information you need to compare their relative sizes. In fact, being circles, pizzas are very tricky in that you may actually have two pizzas side by side and you probably won't be able to tell by how much the area of one is larger than that of the other. On the other hand, if you had one square beside another and the first were four times as large as the second, it would be very clear as in your mind you could actually fit four of the second one perfectly into the first one. That, however, is not how circles are made. You can't fit four circles side by side in a circle that's four times as big - you'd have to, well, melt them and reshape the resulting area.
Okay. This problem, of course, is exactly why pi exists. Pi will allow us to find the truth here where looking at pizzas just won't cut it. There is a formula for this, but look out because it looks a lot for another formula. What we want is the area, and that's radius times pi squared; the other one is for the circumference (which we don't care about), and that's 2 times pi times radius.
So what the pizza place does give me is the diameter, which is twice the radius. There's a 25cm-diameter pie and a 35cm-diameter one, which translates to a radius of 12.5 and another of 17.5. This is what explains the confusion. The difference between 35cm and 25cm (or between their halves) doesn't seem that big. That's because we tend to think about this as if measuring distances with a ruler. But what matters here is the difference between the squares. And that's basically all you need to compare - meaning, since square centimeteres probably won't meaning anything to you (as in, "oh, that look like 500cm2) and pi will stay the same for every pizza, just comparing the sizes of the squares of the radii will give you your answer. 12.5 squared is 156.25 and 17.5 squared is 306.25.
So you see, a pizza whose diameter is only 1.4 as large as another is actually 1.96 as big. And in case you're wondering, yes, 1.96 is the square of 1.4, which gives us perhaps the best way to do all this. Just divide the large diameter by the smaller one, and get the square of that result, and that's by how much the bigger pizza is truly bigger.
As for the actual options I had in this example, they were 20cm, 25, 35 and 40 cm. Let's take the smallest as our pizza-size unit.
25 divided by 20 is 1.25, which squared is 1.5625
35 divided by 20 is 1.75, which squared is 3.0625
40 divided by 20 is 2, which squared is 4
And that's all you need. If you know the size of the small one, now you know that the biggest one is 4 times as big, and the next one is 3 times as big. One very simple and useful thing to keep in mind is that a pizza that sound twice as big is actually 4 times as big (so it would be fair even if it cost 3 times as much)! And that's also about all that you can use. A pizza that sounded 3 times as big as another would be 9 times as big, which seems quite unlikely to come across.
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