Pop Video: How Scale Factor Maths Upset My Pizza Night

Video Transcript

This is the story about how I got banned from our local pizza delivery place. We decided we’d like a pizza for tea one night. And I was nominated to place the call to order a 12-inch deep-pan vegetarian pizza with garlic bread and a bottle of pop.

Unfortunately, they were out of 12-inch bases. So the pizza dude offered me two six-inch pieces for the same price. Now, that sounds reasonable, doesn’t it? Well, actually, I didn’t think so. You see, the six inch and the 12 inch refer to the diameters of the pizzas. So the 12-inch pizza is twice as long in each direction as a six-inch pizza. So twice as long and twice as wide, it’s got four times the area.

Okay, the formula for the area of a circle is 𝜋 times the radius squared. The radius of a six-inch pizza is three inches. That’s half its diameter. So its area is 𝜋 times three squared, which is about 28.3 square inches. The radius of a 12-inch pizza is six inches. So its area is 𝜋 times six squared which is about 113.1 square inches.

Try it on your calculator now. The 12-inch pizza has got four times the area of the six-inch pizza. I should ask for four six-inch pizzas for the price of one 12 inch then. What do you reckon? Well, the thing is, the 12-inch pizza was also twice as deeper as the six-inch pizza. So twice as long, twice as wide, and twice as deep makes two times two times two. That’s eight times the volume. So I asked for eight six-inch pizzas for the price of one 12-inch pizza.

Then, he tried offering me three four-inch pizzas. But that was an even worse deal. The width of a 12-inch pizza is three times that of a four inch. The length is three times and the depth is three times. So the volume of a 12-inch pizza is three times three times three. That’s 27 times as big as a four-inch pizza. I’d need 27 of them to get the same amount of pizza.

But pizza dude wasn’t having any of it. And I tried to give him a short math lesson to prove I was right. But it didn’t go well. In the end, his manager explained to me that he had my name and number. And he knew where I lived. And I would not be bothering them again, making ridiculous demands and upsetting their staff. But I was right!

So if you want to buy a pizza without any stress or worry, you need to forget the following information about length, area, and volume scale factors of similar shapes whose dimensions are all in the same proportions, like small or large pizzas.

If the lengths are twice as big, then the area will be four times as big and the volume will be eight times as big. If the lengths are three times as big, then the area will be nine times as big and the volume will be 27 times as big.

In short, the area scale factor is the square of the length scale factor. And the volume scale factor is the cube of the length scale factor.