The A4 format as a division of a surface of 1 square meter
We all know the A4 format as a rectangle of 297 x 210 mm. But who knows that it is a mathematical construction that implies an A0 format and subsequently A1, A2 and A3?
The basis is the A0 format, a rectangle with a 1 square meter area, with particular proportions. We shall find the ratio of the length \( L_0 \) over width \( W_0\) when we construct the A1 format.
The A1 format is half a A0 format. More precisely, its length \( L_1 \) is the width \(W_0\) of the A0 format, and its width \(W_1\) is half the length \(L_0\) of the A0 sheet.
The characteristic property of the A0 to A1 construction is that the ration of the length to the width is the same for both formats: \(\frac{L_1}{W_1}=\frac{L_0}{W_0}\)
But \( L_1=W_0 \) and \( W_1=\frac{L_0}{2} \), so that: \( \frac{L_1}{W_1} = \frac{W_0}{\frac{L_0}{2}} = 2 \frac{W_0}{L_0} = \frac{2}{\frac{L_0}{W_0}} \).
But \(\frac{L_1}{W_1}=\frac{L_0}{W_0}\), so that \(\frac{L_0}{W_0}=\frac{2}{\frac{L_0}{W_0}}\).
If we multiply both sides by \(\frac{L_0}{W_0}\), we obtain \(\frac{L_0}{W_0}^2=2\), so that \(\frac{L_0}{W_0}=\sqrt{2}\) (\(L_0\) is about \(1.41W_0\)).
But the surface of the A0 format, \(L_0 W_0\), is equal to 1 \(m^2\), or \(100 \times 100=10,000 cm^2\), so that \(\sqrt{2}W_0^2=10,000 cm^2\).
Solving that leads to \(W_0=\sqrt{10,000/\sqrt{2}}=\sqrt{5,000\sqrt{2}}\), and \(L_0=\sqrt{2}W_0=100\sqrt{\sqrt{2}}
\).
This gives \(L_0\) approximately equal to \(118.9 cm\) and L0 is approximately equal to \(84.1 cm\).Thus the A0 format is about \(1189 \times 841 mm\).
For the A1 format, we exchange the length and the width and divide the old length by 2 to obtain the new width: A1 is \(841 \times 595 mm\), with the same proportions as A0 format, as we constructed the A0 format so.
Then we continue: the A2 format has length \(L_2=W_1\) and width \(\frac{L_2}{2}\)) The ratio \(\frac{L_2}{W_2}\) is also \(\sqrt{2}\). A2 format is \(595 \times 420 mm\).
Then A3 format is constructed by dividing the A2 rectangle in 2. The ratio \(\frac{L_33}{W_3}\) is also \(\sqrt{2}\), as it may be shown the same way as for A2 construction from A1 format. A3 format is \(420\times 297 mm\), as it is known.
And now, the famous A4 format, dividing the length of the A3 sheet in 2. The ratio \(\frac{L_4}{W_4}\) is also \(\sqrt{2}\). And the A4 so constructed is \(297 \times 210 mm\), as we already knew!
But now we know where it comes from.
And we know as well why half an A4 is A5 and half a A5 is A6, and so on.
The basis is the A0 format, a rectangle with a 1 square meter area, with particular proportions. We shall find the ratio of the length \( L_0 \) over width \( W_0\) when we construct the A1 format.
The A1 format is half a A0 format. More precisely, its length \( L_1 \) is the width \(W_0\) of the A0 format, and its width \(W_1\) is half the length \(L_0\) of the A0 sheet.
The characteristic property of the A0 to A1 construction is that the ration of the length to the width is the same for both formats: \(\frac{L_1}{W_1}=\frac{L_0}{W_0}\)
But \( L_1=W_0 \) and \( W_1=\frac{L_0}{2} \), so that: \( \frac{L_1}{W_1} = \frac{W_0}{\frac{L_0}{2}} = 2 \frac{W_0}{L_0} = \frac{2}{\frac{L_0}{W_0}} \).
But \(\frac{L_1}{W_1}=\frac{L_0}{W_0}\), so that \(\frac{L_0}{W_0}=\frac{2}{\frac{L_0}{W_0}}\).
If we multiply both sides by \(\frac{L_0}{W_0}\), we obtain \(\frac{L_0}{W_0}^2=2\), so that \(\frac{L_0}{W_0}=\sqrt{2}\) (\(L_0\) is about \(1.41W_0\)).
But the surface of the A0 format, \(L_0 W_0\), is equal to 1 \(m^2\), or \(100 \times 100=10,000 cm^2\), so that \(\sqrt{2}W_0^2=10,000 cm^2\).
Solving that leads to \(W_0=\sqrt{10,000/\sqrt{2}}=\sqrt{5,000\sqrt{2}}\), and \(L_0=\sqrt{2}W_0=100\sqrt{\sqrt{2}}
\).
This gives \(L_0\) approximately equal to \(118.9 cm\) and L0 is approximately equal to \(84.1 cm\).Thus the A0 format is about \(1189 \times 841 mm\).
For the A1 format, we exchange the length and the width and divide the old length by 2 to obtain the new width: A1 is \(841 \times 595 mm\), with the same proportions as A0 format, as we constructed the A0 format so.
Then we continue: the A2 format has length \(L_2=W_1\) and width \(\frac{L_2}{2}\)) The ratio \(\frac{L_2}{W_2}\) is also \(\sqrt{2}\). A2 format is \(595 \times 420 mm\).
Then A3 format is constructed by dividing the A2 rectangle in 2. The ratio \(\frac{L_33}{W_3}\) is also \(\sqrt{2}\), as it may be shown the same way as for A2 construction from A1 format. A3 format is \(420\times 297 mm\), as it is known.
And now, the famous A4 format, dividing the length of the A3 sheet in 2. The ratio \(\frac{L_4}{W_4}\) is also \(\sqrt{2}\). And the A4 so constructed is \(297 \times 210 mm\), as we already knew!
But now we know where it comes from.
And we know as well why half an A4 is A5 and half a A5 is A6, and so on.