Metric paper which always has the same ratio of dimensions when folded in half that being 1/sqrt(2). How was the logic behind this derived?

Question

Metric paper which always has the same ratio of dimensions when folded in half that being 1/sqrt(2). How was the logic behind this derived?

In: Mathematics

Answer 1

You can derive that logic from algebra.

Let’s define a rectangle with its sides being x and y, x<y (so y is the long side). We could define the ratio of the length r = x/y

Here’s the important part: we want this ratio to stay the same when the longer side is folded aka r = (y/2)/x. Why? Because it makes it more practical to resize a A4 sheet into a A5 postcard without cutting parts of what you’re printing.

So by moving terms in both expressions, we want x = r * y AND y = 2 * r * x at the same time. Replace x in the 2nd expression with the 1st, you get y = 2 * r * r * y. Divide both sides by y, you get 1 = 2 * r^2

Now you just have to solve this expression, and you get r = 1/sqrt(2)