Areas and volumes of similar shapes

Last time

Before we start: last time, in Constructions and loci, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

Double the size, how much more?

A pizza has a diameter of ten inches. A pizza twice as wide is not twice the pizza. Its area is four times as big. And a box twice as long, wide and tall holds eight times as much. Lengths, areas and volumes do not scale together, and that surprise is today's lesson.

The three scale factors

For similar shapes, if the length scale factor is k, then areas scale by k squared. Volumes scale by k cubed. With k equal to three, areas are nine times bigger and volumes twenty-seven times bigger. All of this needs the shapes to be similar: the same shape, with every length scaled by the same factor.

Check with a rectangle

A rectangle is four by six, area twenty-four. Double both sides to get eight by twelve. The new area is ninety-six. Ninety-six over twenty-four is four. That is two squared.