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?