Why does enlarging A4 to A3 take 141%, not 200%?
The paper has twice the area, but its height and width are only about 1.414 times as large. Move between a size chart, a folded sheet, and a copier's display to discover the mathematics hidden in A-series paper.
Estimated time for the article text only. Allow extra time for figures, comparisons and interactive exhibits.
Enlarging A4 to A3 doubles the area and ideally multiplies the side lengths by √2. The 141% setting, paper dimensions expressed in whole millimeters, and ideal geometric similarity each have different levels of precision and different roles.
A3 is twice the size of A4. Should you copy at 200%?
You want to enlarge an A4 document to the next size up, A3. Since the paper has twice the area, it is tempting to assume that the enlargement setting should be 200%. Yet some copiers use a preset of 141% for A4 to A3. Canon's official specifications for the TR9530a list that combination. Do "twice the size" and 141% contradict each other?
First, separate what the two numbers describe. The amount of tabletop a sheet covers is its area. How far the printed letters and diagrams stretch vertically and horizontally is a scale factor for length. Both describe how many times larger something becomes, but they compare different quantities.
Take a square with sides 1 centimeter long and double it both vertically and horizontally. Each side is now 2 centimeters long, and four of the original squares fit inside it. The side length has doubled, but the area has quadrupled. That small square leads directly to the number on the button for enlarging A4 to A3.
READ THE COMPARISON / Make a discovery in the diagram
Compare doubling the area with doubling the sides.
At 100% side length, the area is unchanged (×1). At about 141.4%, it is about 2 times the original; at 200%, it is 4 times.
Area scale factor = side scale factor × side scale factor
| Size name | Short side × long side |
|---|---|
| A4 | 210 × 297 mm |
| A3 | 297 × 420 mm |
Comparison conditionsAn original calculated diagram comparing ideal 1:√2 rectangles on the same scale. Actual A-series dimensions are rounded to millimeters. A copier’s 141% indicates the side-length scale factor and differs slightly from exact √2 scaling.
Change the side-length scale factor and compare →See the evidence behind the diagram and the article’s sources
- Japan Paper Association — Paper Q&A: How are paper sizes determined? ↗ (opens in a new tab)
- Ricoh — Paper / Technical information on supplies ↗ (opens in a new tab)
- Canon — TR9530a specifications / Basic copying specifications ↗ (opens in a new tab)
Compare the size chart in two different ways
Ricoh's official paper-size chart lists A4 as 210 × 297 mm and A3 as 297 × 420 mm. Calculating their areas gives 62,370 square millimeters for A4 and 124,740 square millimeters for A3. Divide the latter by the former, and the result is indeed exactly 2. For these two listed sets of dimensions, we can directly confirm that the area doubles.
But comparing the short sides gives 297 ÷ 210, or approximately 1.4143, while comparing the long sides gives 420 ÷ 297, or approximately 1.4141. Neither side doubles. Both dimensions increase by about 1.414, and multiplying the two factors gives an area roughly twice as large. We will return to how dimensions are presented to explain why those two ratios differ slightly.
You can check another way, too. Place two A4 sheets in the same orientation side by side, with their long edges touching. Their outside dimensions are 420 × 297 mm: the size of A3 in landscape orientation. Placing two sheets together and enlarging a diagram within one sheet are different operations. Keeping them distinct is the first step.
How much must the sides grow to double the area?
When you enlarge a diagram while preserving its proportions, you multiply its height and width by the same factor. Call that factor r, and the area increases by r × r, or r squared. To double the area, choose the positive number whose square is 2. That is √2, approximately 1.41421356.
Express the length scale factor as a percentage, and it becomes approximately 141.421%. The preset value of 141% is close to this. The area is 200%; the sides are about 141.4%. Adding the labels "area" and "side length" makes it clear that the numbers describe the same enlargement on different scales.
What happens if you set both dimensions to 200%? A4's 210 × 297 mm becomes 420 × 594 mm, the dimensions listed for A2. You have skipped A3 and moved to paper with four times the area. Does a "double-size copy" mean doubling the side lengths or doubling the area of the paper? One quick clarification makes a substantial difference to the result you should expect.
A shape that stays similar when cut in half
There is a reason the A series has an aspect ratio of 1 to √2. The Japan Paper Association explains that successively halving A0 along its long dimension gives A1, A2, A3, and A4, each with half the area and a geometrically similar shape. Cut the paper smaller and turn it, and its proportions match the original.
Let the short side be a and the long side be b. The ratio of long side to short side in the original rectangle is b ÷ a. After halving the long dimension, the new sheet's short side is b ÷ 2 and its long side is a, giving the ratio a ÷ (b ÷ 2). If the two ratios are equal, b ÷ a = 2a ÷ b. Rearranging shows that the square of b ÷ a is 2, so the ratio must be √2.
Cut a square in half and it becomes an elongated rectangle; it no longer keeps the same shape. The usefulness of the A-series ratio is clearer when we view it through the condition "halve it and turn it, and the shape stays the same" than when we regard it merely as an attractive proportion. This is where folding a sheet of paper meets the mathematics of similar shapes.
Each step up in number halves the paper
The ideal A0 sheet that begins the A series has an area of 1 square meter. Halve it once for A1, twice for A2, three times for A3, and four times for A4. Their ideal areas therefore run 1, 1/2, 1/4, 1/8, and 1/16 of a square meter.
A larger number means smaller paper because the number counts how many times the sheet has been halved. Going back from A4 to A3 reverses one halving, so the area doubles. Going from A4 to A2 reverses two halvings, so the area quadruples. Each step multiplies the side lengths by √2. Two steps give √2 × √2 = 2, matching the 200% setting we considered earlier.
Paper-size numbers can therefore be read as a count of halvings as well as remembered as product names. The fractions listed here, however, describe the ideal relationship that generates the series. Actual size charts give side lengths in whole millimeters, so do not assume that the area of every listed size matches its ideal fraction exactly.
√2, 297 mm, and 141% use different levels of precision
The ideal ratio √2, the whole-millimeter dimensions in a size chart, and the whole-percentage settings provided by a machine are numbers expressed at different levels of precision. Multiply A4's 210 mm by √2 and you get approximately 296.985 mm; multiply its 297 mm by √2 and you get approximately 420.021 mm. Even with the same mathematical scale factor, tiny differences remain from the chart's 297 × 420 mm.
This also explains why the short-side ratio was approximately 1.4143 while the long-side ratio was approximately 1.4141. Ratios calculated using only the listed whole-millimeter dimensions do not exactly reproduce ideal geometric similarity. Distinguishing the practical dimensions shown in the chart from the ideal shapes used to explain the series removes the apparent contradiction.
Furthermore, in a model that simply multiplies the lengths by the displayed 141%, 210 × 297 mm becomes 296.1 × 418.77 mm. This is arithmetic using the specified factor, not a measured print from a real machine. The number 141% alone does not establish that the setting is exactly equal to √2 or that the result will align perfectly with every edge of A3.
The sheet's dimensions and its printable area are different, too
Knowing the outside dimensions of a sheet does not tell us how the edges of the original will be printed. Machines and settings include conditions such as automatic scaling to fit the paper and whether borders are retained. A scale factor calculated from geometry alone cannot determine how the actual page will fit.
The specifications for the same Canon model list automatic scaling, but also explain that borderless copying slightly enlarges the original and does not print the parts extending beyond the paper. This is an example from one particular model, but it shows why filling a sheet without gaps and reproducing the entire original without cropping are not always the same operation.
When enlarging a document, first check the sizes of the original and the output paper, then the scale factor for length, and finally the margins and automatic adjustment settings. Separating those checks makes it easier to tell whether a calculation is wrong or a setting is making an additional adjustment. The mathematics of paper becomes practical when used together with the machine's conditions.
Label what your copying percentage measures
The two numbers that appear when enlarging A4 to A3—twice the size and 141%—describe area and side length respectively. Even when enlarging the same diagram, the scale factor depends on what you measure. Whole-millimeter dimensions and machine settings add further layers, so we also needed to read the ideal √2 separately from the displayed 141%.
Put two sheets side by side to see twice the area. Halve the long dimension to explore the ratio that preserves the shape. Multiply the two numbers in a size chart to get the area; divide them to get the ratio of the sides. A single sheet becomes a small testing ground for comparing the same object through different quantities.
Next time someone asks you to "make it twice as big," consider whether they mean height and width or area. The copier's seemingly odd number 141 is a clue to changing area while preserving shape. In the corner of a familiar A4 sheet, the stories of folding, ratios, and rounding fit neatly together.
Sources and verification notes
This article is an original synthesis of primary sources and explanations from specialist institutions.
Generative AI assists with research, structure and English translation. Claims are compared with sources and editorial checks are performed; independent specialist and native-English reviews have not yet been completed.
- Japan Paper AssociationPaper Q&A: How are paper sizes determined? ↗ (opens in a new tab)
- RicohPaper / Technical information on supplies ↗ (opens in a new tab)
- CanonTR9530a specifications / Basic copying specifications ↗ (opens in a new tab)
Sources checked:
The Japan Paper Association's explanation, Ricoh's size chart, and Canon's official model specifications were opened and checked for geometric similarity and halving, the listed dimensions of A4, A3, and A2, the 141% scaling preset, and the note on borderless copying. The examples of areas, scale factors, and output dimensions are independent arithmetic calculations, not measurements of paper or copiers. This does not constitute a review of rounding methods or dimensional tolerances in the full ISO/JIS standards, or of machines' internal processing.
Rights, status, and revision history
- Status
- Free public edition. Preparation and revision history is retained below.
- 0.1.1
- Rights to the text and diagrams
- The original Japanese text is an independently written summary. The SVG illustrations and conceptual diagrams were newly created. No external images, original source text, or audio recordings are reproduced.
- How sources are used
- Sources are referenced by links only. All rights to each source remain with its respective rights holders.
- · v0.1.0 — Checked primary sources and explanations from specialist institutions to create a private preview draft with original comparisons and diagrams. Connected it to a flagship feature. Formal publication has not been approved.
- · v0.1.1 — Added a reasoned detour to the v17 Earth & Humanity feature. Existing prose, comparison conditions, sources and source-check dates are unchanged.
For another read, another day.
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These paths are editorial comparisons and associations. They do not claim that different subjects work by the same principle.
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