### Mathematics of Paper Sizes

International Organization for Standardization (ISO) standard

Paper sizes are based on the metric system. ISO 216 defines the A series of paper sizes as follows:

• The height divided by the width of all formats is Ö2  or  1.4142. (We take height > width)
• Format A0 has an area of one square meter.
• Format A1 is A0 cut into two equal pieces
• All smaller A series formats are defined in the same way by cutting the next larger format in the series parallel to its shorter side into two equal pieces.
A(n) = 2 A(n+1)
• The standardized height and width of the paper formats is a rounded number in millimeters. Height > width in the above diagram

What is the size of A0 paper?

Let h be the height and w be the width of the A0 paper.

Then    hw = 1000 x 1000   (mm2)

But          h : w = Ö2 : 1

\    h = Ö2 w

\    hw = (Ö2 w)w = 1000 x 1000

\    w2 = 1000000/Ö2 What is the size of A4 paper?

Let h(n) and w(n) be the height and width of A(n) paper.

From the above, we get: But      w(1) = h/2  and  h(1) = w

Therefore      the size of the A1 paper is           (h/2) x w

Similarly,       the size of the A2 paper is            (w/2) x (h/2)

the size of the A3 paper is            (h/4) x (w/2)

the size of the A4 paper is           (w/4) x (h/4) General formulas

The width and height of A(n) paper are given by:

If n is even, If n is odd, Can you verify the table on the left hand side?

 A series paper width x height (mm) A0 841 �� 1189 A1 594 �� 841 A2 420 �� 594 A3 297 �� 420 A4 210 �� 297 A5 148 �� 210 A6 105 �� 148 A7 74 �� 105 A8 52 �� 74 A9 37 �� 52 A10 26 �� 37

B Series paper

The width and height of a B series format is the geometric mean between the corresponding A format and the next larger A format.

Let W(n) and H(n) be the width and height of B(n).  w(n) and h(n) be that of A(n).

Then  B series paper width x height (mm) B1 707 �� 1000 B2 500 �� 707 B3 353 �� 500 B4 250 �� 353 B5 176 �� 250 B6 125 �� 176 B7 88 �� 125 B8 62 �� 88 B9 44 �� 62 B10 31 �� 44

Challenge

1.          Can you check how the entries of the table of B series paper are filled?

2.          Can you get general formulas for W(n) and H(n) of the B(n) paper?

3.          The following magnification factor table appear on photocopying machines :

 Magnification factor Conversion 71% Ö0.5 A3® A4 84% Ö(Ö0.5) B4®A4 119% Ö2 A4®B4 or B5®A4 141% Ö(Ö2) A4®A3 or A5®A4

Remember that two A4 papers have the same size as one A3 paper, and so on.

Do you know how this magnification factor table is constructed?