The sheets we commonly use have a format precise, theA4which 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 formatsrespectively half And double of the A4.
This type of format was defined in 1922 starting from the A0 sheetmaintaining 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 theA4which has the following dimensions 21cm x 29.7cmis 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 126was established by a convention adopted in Germany in 1922 from the Deutsches Institut für Normung and started from A0 sheeta sheet of 1 m2 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 theA0, 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, √2ensures 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 BThe 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 sheetthe proportion between the sides comes maintained, and this happens for any value of TO And B such that their ratio is √2.
