![]() The medium-size cracker has length B/ A times the length of the small one, so its area is scaled by ( B/ A) 2, and so its area is kB 2. This is because the small cracker has area proportional to A 2, let’s call its area kA 2 (for some scaling factor, k). They don’t make crackers in this shape, but if I baked my own, I’d know that the areas work out. ![]() In the example above, the blue triangle is a right triangle and the three squiggly shapes are similar, so the area of the red one is equal to the sum of the areas of the two green ones. It is easy to prove that the area of the large one is equal to the sum of the areas of the smaller ones. So in general one can choose any plane shape and make three similar copies of it scaled to fit on the sides of a right triangle. (I’m happy to stick with the traditional term “squared.”) We say nine is “three squared” but the picture shows that we could just as well say nine is “three triangled.” Or, in fact, we could say “three pentagoned.” It is true for any plane figure that if you scale its length by n then the area is scaled by n 2. This squaring relationship between length and area is not specific to squares. ![]() ![]() The first thing to notice is that the triangle of edge length three contains nine crackers, the triangle of edge length four contains sixteen crackers, and the triangle of edge length five contains twenty five crackers. ![]()
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