Why A4 Paper Hides a Square Root, math that surprises you lesson

Cut an A4 sheet in half across its long side and each half is an A5 sheet, which looks like a smaller copy of the A4 sheet. Most rectangles do not behave like this. Halve a rectangle that is twice as long as it is wide and you get two squares. Which rectangle keeps exactly its shape when it is cut in half? This page shows that its long side must be $\sqrt{2}$ times its short side, builds the whole A series from that one ratio, and compares the ideal sizes with the rounded millimeter sizes of real paper.

Key terms

A-series paper sizes

Definition The A-series paper sizes are a family of rectangular sheet sizes defined in the international standard ISO 216, which goes back to the German standard DIN 476 of 1922. In the ideal version, with exact lengths, the largest size A0 has area exactly $1$ square meter and a long side $\sqrt{2}$ times its short side. Each later size is one of the two halves made by cutting the size before it along the line that joins the midpoints of its long sides, so A1 is half of A0, A2 is half of A1, and so on. The ratio $\sqrt{2}$ is the only one for which such a half has the same shape as the whole sheet. So every ideal size has the same shape, and the size $\text{A}n$ has area $1/2^n$ of a square meter and sides

\begin{gathered} 2^{-1/4 - n/2}\ \text{m} \\[0.4em] \text{and} \quad 2^{1/4 - n/2}\ \text{m}. \end{gathered}

The standard itself lists every size in whole millimeters. A0 is $841$ by $1189$ millimeters, its ideal sides rounded to the nearest millimeter. Each later size takes the short side of the size before it as its long side, and half of that size’s long side, with any half millimeter dropped, as its short side.

In plain words One rule makes the whole family. Start with a square meter of paper in the one shape that survives halving, and keep cutting it in half. The number after the A counts the cuts.

Example The first five sizes are these, with all lengths in millimeters.

Size	Ideal area in square meters	Ideal sides	Nominal sides
A0	$1$	$840.9 \times 1189.2$	$841 \times 1189$
A1	$1/2$	$594.6 \times 840.9$	$594 \times 841$
A2	$1/4$	$420.4 \times 594.6$	$420 \times 594$
A3	$1/8$	$297.3 \times 420.4$	$297 \times 420$
A4	$1/16$	$210.2 \times 297.3$	$210 \times 297$

Halving the nominal long side of A0 gives $1189/2 = 594.5$ , and dropping the half millimeter gives the $594$ of A1. Because half millimeters are dropped, two sheets of one nominal size never take up more room than one sheet of the size before it. Four cuts lead from A0 to A4, so an ideal A4 sheet has area $1/16$ of a square meter, which is $62{,}500$ square millimeters.

A4 paper

Definition A4 paper is the A-series size four halvings below A0. As an ideal rectangle it has area $1/16$ of a square meter, or $62{,}500$ square millimeters, and aspect ratio exactly $\sqrt{2}$ , with sides of about $210.22$ and $297.30$ millimeters. Its nominal size in the standard is $210$ by $297$ millimeters, and a manufactured sheet may differ from that by a small tolerance that the standard allows. Since $210$ and $297$ are whole numbers, the nominal ratio $297/210$ is a fraction of whole numbers, and no such fraction equals $\sqrt{2}$ . So a nominal A4 sheet is very close to the ideal shape but not exactly that shape.

In plain words An A4 sheet is a sixteenth of a square meter cut to the $\sqrt{2}$ shape and then rounded to whole millimeters. The rounding is far too small to see, but it is there.

Example Dividing both numbers by $3$ gives

\frac{297}{210} = \frac{99}{70} = 1.41428\ldots,

while $\sqrt{2} = 1.41421\ldots$ , so the nominal ratio is larger by about $0.00007$ . On a sheet $210$ millimeters wide, the ideal long side would be $210\sqrt{2} \approx 296.985$ millimeters, about $0.015$ millimeters shorter than $297$ . The nominal area is $210 \cdot 297 = 62{,}370$ square millimeters, which is $130$ less than the ideal $62{,}500$ , a difference of about $0.2$ percent. Both small gaps come from using whole millimeters. The halving argument for the ideal shape is exact.

Aspect ratio

Definition On this page the aspect ratio of a rectangle is its long side divided by its short side. It is at least $1$ , it equals $1$ only for a square, and it does not change when the rectangle is turned or measured in another unit. Take a rectangle with long side $L$ , short side $S$ and aspect ratio $r = L/S$ , and cut it along the line through the midpoints of its long sides. Each half has sides $S$ and $L/2$ . When $r < 2$ , the side $S$ is the longer one, and each half has aspect ratio

\frac{S}{L/2} = \frac{2S}{L} = \frac{2}{r}.

When $r = 2$ the halves are squares, and when $r > 2$ their aspect ratio is $r/2$ .

In plain words Halving a sheet across its long side swaps which side is long. As long as the old aspect ratio is less than $2$ , the new one is $2$ divided by the old one.

Example A sheet keeps its shape under halving exactly when each half has the same aspect ratio $r$ as the sheet. For $r > 2$ the half has aspect ratio $r/2$ , which is smaller than $r$ , and for $r = 2$ it has aspect ratio $1$ , so neither case works. For $1 \le r < 2$ the condition is

r = \frac{2}{r}, \quad \text{so} \quad r^2 = 2.

The negative root $-\sqrt{2}$ cannot be a ratio of lengths, and the positive root $\sqrt{2} \approx 1.414$ does lie between $1$ and $2$ , so $r = \sqrt{2}$ is the one answer. Other rectangles change shape at every cut. A rectangle with aspect ratio $2$ halves into squares, and a square halves into rectangles with aspect ratio $2$ , so these two shapes take turns. A sheet of US letter paper, $8.5$ by $11$ inches, has aspect ratio $22/17 \approx 1.29$ , and its halves, $5.5$ by $8.5$ inches, have aspect ratio $17/11 \approx 1.55$ . Cutting the other way, through the midpoints of the short sides, never keeps the shape, because those halves have aspect ratio $2r$ . Cutting a sheet into four quarters by halving both of its sides gives pieces with sides $L/2$ and $S/2$ , the original shape for every rectangle, so the surprise is that a single cut can be enough.

Square root of 2

Definition The square root of $2$ , written $\sqrt{2}$ , is the positive number whose square is $2$ . It is about $1.41421356$ , and its reciprocal is $1/\sqrt{2} = \sqrt{2}/2 \approx 0.70711$ . It is irrational, which means it is not equal to any fraction $p/q$ of whole numbers. To see why, suppose that $\sqrt{2} = p/q$ with the fraction in lowest terms. Then $p^2 = 2q^2$ , so $p^2$ is even, and therefore $p$ is even, since the square of an odd number is odd. Writing $p = 2m$ gives $4m^2 = 2q^2$ , so $q^2 = 2m^2$ and $q$ is even as well. A fraction in lowest terms cannot have an even numerator and an even denominator, so no such fraction exists.

In plain words The number $\sqrt{2}$ is the length of the diagonal of a square with side $1$ . Fractions can come as close to it as you like, but none of them hits it exactly, so a rectangle whose sides are whole numbers of millimeters never has aspect ratio exactly $\sqrt{2}$ .

Example The fractions $3/2$ , $7/5$ , $17/12$ , $41/29$ and $99/70$ come closer and closer to $\sqrt{2}$ . Each fraction $p/q$ in this list leads to the next one, $(p + 2q)/(p + q)$ , and the square of each misses $2$ by exactly $1/q^2$ , alternately from above and from below. For the nominal A4 ratio,

\begin{aligned} \left(\frac{99}{70}\right)^2 &= \frac{9801}{4900} \\[0.4em] &= 2 + \frac{1}{4900}. \end{aligned}

So $99/70$ is a little more than $\sqrt{2}$ , by about $0.00007$ , and $41/29$ is a little less, by about $0.0004$ .

Similar rectangles

Definition Two rectangles are similar when one is a scaled copy of the other, possibly turned. That means there is a positive number $k$ , the scale factor, such that the long and short sides of one are $k$ times the long and short sides of the other. Two rectangles are similar exactly when they have the same aspect ratio, because if $L'/S' = L/S$ and $k = S'/S$ , then $L' = kL$ as well. Scaling by $k$ multiplies every length by $k$ and every area by $k^2$ , since both sides of a rectangle are multiplied by $k$ .

In plain words Similar rectangles have the same shape and may differ in size. The aspect ratio measures the shape and the area measures the size, and a cut in half can keep the first while it halves the second.

Example The halves of an ideal A-series sheet are similar to it and have half its area, so $k^2 = 1/2$ and

k = \frac{1}{\sqrt{2}} \approx 0.7071.

Each side shrinks to about $70.7$ percent, and the area to half, since $\tfrac{1}{\sqrt{2}} \cdot \tfrac{1}{\sqrt{2}} = \tfrac12$ . Enlarging A4 to A3 goes the other way, with $k = \sqrt{2}$ , about $141$ percent. Similar rectangles can also be recognized by their diagonals. Lay an ideal A5 sheet on an ideal A4 sheet, both with the long side vertical, so that they share one corner and the two edges at that corner. The diagonal of the A4 sheet from the shared corner then passes exactly through the opposite corner of the A5 sheet, because both diagonals rise $\sqrt{2}$ units for each unit across.

Check yourself

Two more halvings take A4 to A6. Find the ideal area and the ideal sides of an A6 sheet, and compare them with the nominal size $105$ by $148$ millimeters.

Show the answer

Each halving halves the area, so the ideal A6 area is

\frac{62{,}500}{4} = 15{,}625

square millimeters, which is $1/64$ of a square meter. Each halving also multiplies both sides by $1/\sqrt{2}$ , so two halvings multiply them by $\tfrac12$ . The ideal A6 sides are half the ideal A4 sides, about $105.11$ and $148.65$ millimeters. The nominal sizes follow the rounding rule. Half of $297$ is $148.5$ , so A5 is $148$ by $210$ , and half of $210$ is $105$ , so A6 is $105$ by $148$ . The nominal area is $105 \cdot 148 = 15{,}540$ square millimeters, $85$ less than the ideal area. The nominal ratio $148/105 \approx 1.4095$ is further from $\sqrt{2}$ than the A4 ratio, because a half millimeter was dropped on the way.

A copier is set to enlarge an A4 page so that it exactly fills an A3 sheet. Many people guess that this takes $200$ percent. What is the right setting, and what would $200$ percent give?

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An A3 sheet has the same shape as an A4 sheet and twice its area. The scale factor $k$ therefore satisfies $k^2 = 2$ , so $k = \sqrt{2} \approx 1.414$ , a setting of about $141$ percent. At $200$ percent both sides double, so the area grows by a factor of $4$ . An ideal A4 page would become about $420.4$ by $594.6$ millimeters, the size of A2, two steps up. The same mistake happens in reverse. Two A4 pages side by side, sharing a long side, form an A3 shape, so printing them together on one A4 sheet takes $1/\sqrt{2}$ , about $71$ percent, not $50$ percent.

Change the rule. Instead of cutting a sheet in half, cut it into three equal strips by two cuts parallel to its short side. Which aspect ratio makes every strip the same shape as the sheet? What happens with $n$ strips?

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With long side $L$ , short side $S$ and $r = L/S$ , each strip has sides $S$ and $L/3$ . When $r < 3$ , the side $S$ is the longer one and the strip has aspect ratio

\frac{S}{L/3} = \frac{3}{r}.

Setting $3/r = r$ gives $r^2 = 3$ , so $r = \sqrt{3} \approx 1.732$ , which is less than $3$ as needed. When $r = 3$ the strips are squares, and when $r > 3$ they have aspect ratio $r/3$ , smaller than $r$ , so $\sqrt{3}$ is the only answer. With $n$ equal strips the same steps give $r = \sqrt{n}$ . The A series is the case $n = 2$ .