Is there a size of rectangle that retains its ratio when it's folded in half?

Solution 1:

The $1:\sqrt{2}$ ratio ensures exactly that. That is the idea behind the ISO 216 standard for paper sizes, which was adopted from the German DIN 476 standard.

Its most common usage is the A series which especially in Europe is a collection of very common paper sizes. The base size, A0, has an area of a square meter, and every next smaller paper size is constructed by folding it in half.

Solution 2:

A ratio of $1:\sqrt{2}$ will do the trick!

The original rectangle will have a ratio of $x:y$, where $y$ is the longer side and $x$ the shorter side. Then the folded rectangle will have a ratio of $\frac{1}{2}y:x$ and we want

$$\begin{align} \frac{x}{y} & =\frac{\frac{1}{2}y}{x} \\ x^2 & = \tfrac{1}{2}y^2 \\ x & = \tfrac{1}{\sqrt{2}} \cdot y \\ y & = \sqrt{2}\cdot x \end{align}$$

Solution 3:

Well, suppose you have a rectangle of sides with $A$ and $B$ of length, $A$ being the bigger side.

When we fold it along the longest side we end up with a rectangle with sides of length $B$ and $A/2$.

So what you want to know is if there are any values of A and B that satisfy the following condition:

$$\frac{A}{B} = \frac{B}{A/2}$$

From solving the equation we get that $A = B \cdot \sqrt2$. So if the paper's height is $\sqrt2$ times its width, then we can make a rectangle with the properties that you wanted. And this is the only ratio that will work.

Solution 4:

$H:W = W:(H/2)$ resolves to $H:W = \sqrt 2$

Solution 5:

Standard Bond paper sizes A4, A3, A2 and A1 have been designed so that areas double up and sides increase by scale $ \sqrt 2 $.

$$ \frac LB = \frac 12 \cdot \frac bl $$