Paper dimensions look arbitrary until one simple requirement is imposed: cutting a sheet in half should produce two smaller sheets with exactly the same shape. That requirement determines the aspect ratio uniquely. It is why an A4 sheet measures 210 mm by 297 mm rather than some pair of rounder-looking numbers, and why enlarging A4 to A3 uses a scale factor close to 141%.
The Self-Similarity Requirement
Let the long side of a rectangular sheet be L and the short side be S, with \(L>S>0\). Cutting perpendicular to the long side produces two rectangles whose side lengths are \(S\) and \(L/2\). The half-sheet is rotated in the diagram so its orientation matches the original:
Similarity means that the ratio of long side to short side must be unchanged. Before the cut it is \(L/S\); after rotating a half-sheet it is \(S/(L/2)\). Equating them gives
\[ \frac{L}{S}=\frac{S}{L/2}=\frac{2S}{L}. \]
Multiplying by \(L/S\) reduces the condition to
\[ \left(\frac{L}{S}\right)^2=2. \]
Lengths are positive, so only the positive square root is geometrically meaningful:
\[ \boxed{\frac{L}{S}=\sqrt2}. \]
This is not merely a convenient ratio. It is the only positive aspect ratio for which halving the long side and rotating the result reproduces the original shape.
From the Ratio to the A-Series
The ratio fixes the shape but not the absolute size. The A-series adds a second condition: the ideal A0 sheet has area \(1\text{ m}^2\). If its short side is \(S_0\), then its long side is \(L_0=\sqrt2S_0\), so
\[ S_0L_0=\sqrt2S_0^2=1\text{ m}^2. \]
Solving for both sides gives the ideal, unrounded dimensions
\[ S_0=2^{-1/4}\text{ m}\approx0.840896\text{ m}, \qquad L_0=2^{1/4}\text{ m}\approx1.189207\text{ m}. \]
Each step from \(A_n\) to \(A_{n+1}\) halves the area. Therefore
\[ \operatorname{area}(A_n)=2^{-n}\text{ m}^2. \]
Combining that area with the constant ratio \(L_n/S_n=\sqrt2\) gives a closed form for every ideal A-size:
\[ \boxed{S_n=2^{-n/2-1/4}\text{ m}}, \qquad \boxed{L_n=2^{-n/2+1/4}\text{ m}}. \]
Why A4 Is 210 mm by 297 mm
A4 is four halvings below A0, so its ideal area is \(2^{-4}=1/16\) square metre. Substituting \(n=4\) into the formulas above gives
\[ S_4=2^{-9/4}\text{ m}\approx210.224\text{ mm}, \qquad L_4=2^{-7/4}\text{ m}\approx297.302\text{ mm}. \]
ISO 216 specifies practical trimmed dimensions in whole millimetres. The nominal A4 size is therefore 210 mm by 297 mm. Those integers do not have an exact ratio of \(\sqrt2\), nor is their product exactly \(1/16\text{ m}^2\); they are the standardized physical dimensions that approximate the ideal self-similar geometry while keeping successive sizes compatible.
| Size | Nominal dimensions | Ideal area |
|---|---|---|
| A0 | 841 mm × 1189 mm | \(1\text{ m}^2\) |
| A1 | 594 mm × 841 mm | \(1/2\text{ m}^2\) |
| A2 | 420 mm × 594 mm | \(1/4\text{ m}^2\) |
| A3 | 297 mm × 420 mm | \(1/8\text{ m}^2\) |
| A4 | 210 mm × 297 mm | \(1/16\text{ m}^2\) |
| A5 | 148 mm × 210 mm | \(1/32\text{ m}^2\) |
| A6 | 105 mm × 148 mm | \(1/64\text{ m}^2\) |
The distinction between ideal and nominal dimensions matters. Ideal A0 has exactly one square metre of area, but the nominal 841 mm by 1189 mm rectangle has area
\[ 0.841\cdot1.189\text{ m}^2=0.999949\text{ m}^2. \]
The difference is only \(51\text{ mm}^2\), but it explains why treating the printed millimetre dimensions as exact algebraic values eventually produces small discrepancies.
Scaling Between Paper Sizes
Halving the area does not halve each side. Similar rectangles scale in area by the square of their linear scale factor. If \(k\) maps \(A_n\) to \(A_{n+1}\), then
\[ k^2=\frac12, \qquad k=\frac{1}{\sqrt2}\approx0.7071. \]
Reducing one A-size to the next therefore uses about 70.71%. Enlarging reverses the factor:
\[ \frac{1}{k}=\sqrt2\approx1.4142. \]
This is the origin of the familiar 71% reduction and 141% enlargement settings on photocopiers. The displayed percentages are rounded; the exact linear factors are \(1/\sqrt2\) and \(\sqrt2\).
Dividing a Rectangle into More Than Two Parts
The same argument works for \(n\) equal strips. Start with a rectangle whose long-to-short ratio is \(r=L/S\), and divide its long side into \(n\) equal segments. Each strip has sides \(S\) and \(L/n\). After rotating a strip, self-similarity requires
\[ \frac{L}{S}=\frac{S}{L/n}=\frac{nS}{L}. \]
Thus \(r^2=n\), and the unique positive ratio is
\[ \boxed{r=\sqrt n}. \]
For \(n=2\), this returns the A-series ratio \(1:\sqrt2\). Dividing into thirds requires \(1:\sqrt3\), while quartering requires \(1:2\). The square-root rule is therefore not peculiar to paper; it is the general condition for a rectangle to reproduce its shape after equal subdivision along its long side.
What the Ratio Achieves
- Cutting or folding along the midpoint preserves the page shape after rotation.
- Two adjacent sheets of one size approximate one sheet of the preceding size without wasted area.
- Documents can move between adjacent sizes with one fixed linear scale factor.
- Page layouts, margins, images, and type retain their relative proportions after scaling.
The ratio does not make the whole-millimetre dimensions mathematically exact. Physical trimming tolerances and integer dimensions remain part of the standard. The elegance lies in separating the ideal geometry from its practical realization: \(\sqrt2\) determines the shape, one square metre anchors A0, repeated halving determines the series, and standardized millimetre values make it manufacturable.