The sheets we commonly use have a **format** precise, the**A4**which has dimensions of **21cm x 29.7cm***. *This measure may seem random or counterintuitive, but in reality it hides a **specific geometric property**: by folding an A4 sheet in two halves along the longer side, you obtain two sheets that have half the area but **they maintain the proportions perfectly** of the starting sheet. The proportion of an A4 sheet is **the only one with this property**. This proportion is maintained even if we double an A4 sheet by placing it next to another: we obtain a sheet with double the area, but which perfectly maintains the proportions of the original A4 sheet. The ones we have just described are precisely the **A5 and A3 formats**respectively **half** And **double** of the A4.

This type of format was defined in 1922 starting from the **A0 sheet**maintaining the proportions.

But how is it geometrically possible? We will see it in this article.

## The international standard ISO 216: different but proportionally equal formats

The most used paper format is definitely the**A4**which has the following dimensions **21cm x 29.7cm**is part of a specific series of paper formats – **the series A** – established by the International Organization for Standardization in order to define common technical standards worldwide.

The standard for sheets, called **ISO 126**was established by a convention adopted in Germany in **1922** from the *Deutsches Institut für Normung* and started from **A0 sheet**a sheet of **1 m ^{2} **whose size of the

**sides**(841 mm x 1189 mm) was established so that their

**relationship**was equal to

**√2**

*.*

It can be shown that this specification **proportion** between the sides (**1 : √2**) means that by halving the sheet along the longer side, two sheets are obtained with **half of the area** but which maintain the **same exact proportions** of the starting sheet.

In this way, **starting from the****A0***, *you get the formats **A1, A2, A3, A4, A5 **and so on, having halved area from format to format, but maintaining the same proportions. **Doing** Indeed **the relationship** between the dimensions of the sides of a sheet *A4 *you get **29.7/21 ≃ 1.4142 ≃ √2***, *and the same happens when calculating the ratio between the sides of the other formats.

The fact that the proportion between the sides is maintained when switching from format to format has a **extremely practical side**: allows you to** keep images perfectly proportioned** if you switch from one format to another. This is why, for example, in printers we can choose whether to print in A3, A4 or with a double sheet over the A4 format, that is in A5: whatever the format chosen, we are sure that what we will print will not be deformed.

**But how come?** this specific value of the ratio, **√2**ensures that the aspect ratio is maintained when halving an aspect ratio?

## The √2 ratio allows you to maintain the proportion between one format and another

To understand why a rectangle – that is, a sheet of paper – which has a ratio between its sides equal to √2, maintains its proportions if halved, let’s consider a **rectangle** any and let’s call its sides *TO* And *B*The dimensions of *TO* And *B* they are such that their **relationship TO/B** it’s exactly the same as

**√2**

*.*

Now **let’s halve** the sheet along the longer side, thus obtaining two rectangles that have area equal to half of the starting one, and sides equal to ** B And TO/2**. If we now calculate the ratio between the two sides, we get:

So, **halving the sheet**the **proportion** between the sides comes **maintained,** and this happens for any value of *TO* And *B* such that their ratio is √2.