# Paper size system and the ideas behind its design

A4 is the king of papers and widely used all over the world today. Why the king size is not the perfect 200x300 mm!?

1. Paper size design ideas
2. Divide the sheet by folding
3. Paper aspect ratio
4. A series
5. B Series
6. C Series
7. Sizes of drawings

Many paper size standards conventions have existed at different times and in different countries. Today there is one widespread international ISO standard.

## Paper size design ideas

• Each paper size is one half of the area of the next size up.
• Width -to- Height ratio of all pages is constant. w/h = constant
• Grandmother A0 size has an area of one square meter. $A0=1{m}^{2}$

The letter, written in 1786-10-25 by the physics professor Georg Christoph Lichtenberg (1742–1799) to Johann Beckmann, seems to be the oldest preserved written reference to the idea of using the square-root of two as an aspect ratio for paper formats:

. . . I once gave an exercise to a young Englishman, whom I taught in algebra, to find a sheet of paper for which all formats are similar to each other. Having found that ratio, I wanted to apply it to an available sheet of ordinary writing paper with scissors, but found with pleasure, that it already had it. It is the paper on which I write this letter, . . .

Lichtenberg

## Divide the sheet by folding

Successive paper sizes in the series  A0, A1, A2, A3, A4, and so forth, are defined by halving the preceding paper size along the larger dimension. Each format is cut into two equal pieces. In other words, the height of A1 is the width of A0 and the width of A1 is half the height of A0.

Areas: A0=2A1, A1=2A2, A2=2A3, A3=2A4

No waste: A0=A1 +A2 +A3 +A4 +A5 +A6 +A7 +2A8

## Paper aspect ratio

### ANSI standard sizes have two aspect ratios

Unlike the ISO standard, however, the arbitrary aspect ratio forces this series to have two alternating aspect ratios. The ANSI series is shown below.

ANSI standard sizes have two alternating aspect ratios
SizeWidth x HeightAspect Ratio
A8.5" x 11.0"1/1.294
B 11.0" x 17.0"1/1.545
C17.0" x 22.0"1/1.294
D22.0" x 34.0"1/1.545
E34.0" x 44.0"1/1.294

ANSI (American National Standards Institute) defined a regular series of paper sizes based around the Letter (8.5" x 11") format, with this becoming the A sizes and larger sizes being B , C , D , and E. Surprisingly these ANSI standard sizes were defined in 1995 well after the ISO standard sizes.

Unlike the ISO standard sizes which have the single aspect ratio of $\frac{1}{\sqrt{2}}$, ANSI standard sizes have two aspect ratios $\frac{1}{1.294}$ and $\frac{1}{1.545}$ which means that enlarging and reducing between the sizes is not as easy as with the ISO sizes and leaves wider margins on the enlarged/reduced document.

### Nice round paper sizes have not single aspect ratios

Unlike the ISO standard, however, the arbitrary aspect ratio forces this series to have two alternating aspect ratios 2/3 and 3/4. My favourite imaginery round paper sizes have two aspect ratios. Our dream of constant aspect ratio haven't come true.

My favourite imaginary round paper sizes have two alternating aspect ratios
My favourite imaginery paper sizesWidth x HeightAspect Ratio
Imaginary size 0800x1200 mm2/3
Imaginary size 1600x800 mm3/4
Imaginary size 2400x600 mm2/3
Imaginary size 3300x400 mm3/4
Imaginary size 4200x300 mm2/3
Imaginary size 5150x200 mm3/4

### Lichtenberg's idea: In the search of a constant aspect ratio

The main advantage of ISO system is its scaling: if a sheet with an aspect ratio of $\sqrt{2}$ is divided into two equal halves parallel to its shortest sides, then the halves will again have an aspect ratio of $\sqrt{2}$ . Folded brochures of any size can be made by using sheets of the next larger size, e.g. A4 sheets are folded to make A5 brochures. The system allo ws scaling without compromising the aspect ratio from one size to another – as provided by office photocopiers, e.g. enlarging A4 to A3 or reducing A3 to A4. Similarly, two sheets of A4 can be scaled down to fit exactly one A4 sheet without any cutoff or margins.

Finding the constant number for aspect ratio is easy: Let a and b be the long side and the short side of the paper respectively.

$ab= ba2=constant$

$a2=2b2$

$ab=2$

The constant aspect ratio is unique.

The geometric rationale behind the square root of 2 is to maintain the aspect ratio of each subsequent rectangle after cutting or folding an A series sheet in half, perpendicular to the larger side.

The advantages of basing a paper size upon an aspect ratio of $\sqrt{2}$ were first noted in 1786 by the German scientist and philosopher Georg Christoph Lichtenberg. Early in the 20th century, Dr Walter Porstmann turned Lichtenberg's idea into a proper system of different paper sizes. Porstmann's system was introduced as a DIN standard (DIN 476) in Germany in 1922, replacing a vast variety of other paper formats. Even today the paper sizes are called "DIN A4" (pronounced: "deen-ah-fear") in everyday use in Germany and Austria. The term Lichtenberg ratio has recently been proposed for this paper aspect ratio.

## A series paper sizes

We can produce infinite series of papers using the constant $\sqrt{2}$ aspect ratio. The base size can be based on rationale $A0=1{m}^{2}$ area.

ISO A series paper sizes
SizeWidth x HeightAreaAspect Ratio
A0 841 × 1189 mm
1 m2 $\frac{1}{\sqrt{2}}$
A1594 × 841 mm1/2 A0$\frac{1}{\sqrt{2}}$
A2420 × 594 mm1/4 A0$\frac{1}{\sqrt{2}}$
A3297 × 420 mm1/8 A0 $\frac{1}{\sqrt{2}}$
A4210 × 297 mm
1/16 A0$\frac{1}{\sqrt{2}}$
A5148.5 × 210 mm 1/32 A0$\frac{1}{\sqrt{2}}$
A6105 × 148.5 mm1/64 A0$\frac{1}{\sqrt{2}}$
A774 × 105 mm1/128 A0$\frac{1}{\sqrt{2}}$
A8 52 × 74 mm1/256 A0$\frac{1}{\sqrt{2}}$
A937 × 52 mm1/512 A0$\frac{1}{\sqrt{2}}$
A1026 × 37 mm1/1024 A0$\frac{1}{\sqrt{2}}$

The base A0 size of paper is defined to have an area of $A0=1{m}^{2}$. Rounded to millimeters, the A0 paper size is 841 by 1,189 millimeters. The weight of each sheet is also easy to calculate given the basis weight in grams per square meter (g/m2). Since an A0 sheet has an area of 1 m2, its weight in grams is the same as its basis weight in g/m2. A standard A4 sheet made from 80 g/m2 paper weighs 5 g, as it is one 16th (four halvings) of an A0 page. Thus the weight, and the associated postage rate, can be easily calculated by counting the number of sheets used.

According to some theorists, A series sizes are generally too tall and narrow for book production. European book publishers typically use metricated traditional page sizes for book production.

## B series paper sizes

B0 sheet is 1 metre wide, and has an area of 1.414 m2
SizeWidth x HeightAreaAspect Ratio
B0 1000 × 1414 mm
1414 m2$\frac{1}{\sqrt{2}}$
B1707 × 1000 mm1/2 B0$\frac{1}{\sqrt{2}}$
B2500 × 707 mm1/4 B0 $\frac{1}{\sqrt{2}}$
B3353 × 500 mm1/8 B0 $\frac{1}{\sqrt{2}}$
B4250 × 353 mm
1/16 B0 $\frac{1}{\sqrt{2}}$
B5176 × 250 mm 1/32 B0 $\frac{1}{\sqrt{2}}$
B6125 × 176 mm1/64 B0 $\frac{1}{\sqrt{2}}$
B788 × 125 mm1/128 B0 $\frac{1}{\sqrt{2}}$
B8 62 × 88 mm1/256 B0 $\frac{1}{\sqrt{2}}$
B944 × 62 mm1/512 B0 $\frac{1}{\sqrt{2}}$
B1031 × 44 mm1/1024 B0 $\frac{1}{\sqrt{2}}$

In addition to the A series, there is a less common B series. The B series was designed to fill the voids between the A series sizes. The area of B series sheets is the geometric mean of successive A series sheets. So, B1 is between A0 and A1 in size, with an area of $\frac{1}{\sqrt{2}}=0.707{m}^{2}$. As a result, B0 is 1 metre wide, and other sizes in the B series are a half, a quarter or further fractions of a metre wide. While less common in office use, it is used for a variety of special situations. Many posters use B-series paper or a close approximation, such as 50 cm × 70 cm; B5 is a relatively common choice for books. The B series is also used for envelopes and passports. The B-series is widely used in the printing industry to describe both paper sizes and printing press sizes, including digital presses. B3 paper is used to print two US letter or A4 pages side by side using imposition; four pages would be printed on B2, eight on B1, etc.

## C series paper sizes

The C series is the standard series for envelopes and folders. Each C size envelope is designed to fit the corresponding A size flat sheet. For example, the C4 envelope is designed for the A4 letter. C4 is slightly larger than A4, and B4 slightly larger than C4. The practical usage of this is that a letter written on A4 paper fits inside a C4 envelope, and C4 paper fits inside a B4 envelope.

## Sizes of drawings

Sizes of drawings typically comply with either of two different standards, ISO(World Standard) or ANSI/ASME Y14 (American), according to the following table.

As stated in the requirements of BS EN ISO 5457, the original drawing should be made on the smallest sheet, permitting the necessary clarity and resolution.

Sizes A0 to A3 shall be used in landscape orientation only, and the location of the title block shall be situated in the bottom right-hand corner of the drawing space.

A4 sheets may be used in landscape or portrait orientation.

A4 A 210 X 297 mm 8.5" x 11" 297 X 420 mm 11" X 17" 420 X 594 mm 17" X 22" 594 X 841 mm 22" X 34" 841 X 1189 mm 34" X 44"

• References:
• ANSI/ASME Y14.1
• ISO 216
• Lichtenberg’s letter to Johann Beckmann
• Wikipedia