A Series Paper Sizes and Their Properties

Overview

A-series paper sizes, established by ISO 216, are widely used internationally due to their unique properties and practical design. The series is based on a consistent aspect ratio of √2 (approximately 1.414), ensuring that each subsequent size maintains the same proportions when folded or cut in half.

Key Properties of the A-Series

• Aspect Ratio:

  The defining feature of A-series paper is its √2 aspect ratio (1:1.414). This ratio ensures that the shape of the paper remains consistent when scaled up or down.

• Base Size (A0):

  The A0 paper size is designed to have an area of exactly 1 square meter (1 m²). This serves as the starting point for the entire series.

• Halving Rule:

  Each successive size in the A-series (A1, A2, A3, etc.) is obtained by halving the area of the preceding size while maintaining the aspect ratio. For example, the short side of A0 becomes the long side of A1.

• Most Popular Size (A4):

  A4, measuring 210 mm × 297 mm (8.27 in × 11.7 in), is the most commonly used size globally. It is equivalent to approximately 1/16th of a square meter (0.0625 m²).

Dimensions of A-Series Paper Sizes

Geometric Rationale

The √2 aspect ratio is critical for maintaining proportionality across the series. When an A-series sheet is folded in half, the resulting dimensions retain the original aspect ratio. This consistency is beneficial for various applications, such as scaled photocopying, where reducing or enlarging documents preserves their layout.

Mathematical Derivation

For a rectangle with dimensions:

• Length = x
• Width = y

The √2 ratio ensures:

x / y = y / (x/2)

Solving for x/y gives x/y = √2.

Comparison with Other Standards

• North American Letter Size:

  Common in the United States and Canada, letter paper measures 8.5 in × 11 in (216 mm × 279 mm). It is slightly wider and shorter than A4.

• Japanese Standards:

  Japan uses a modified A-series along with a unique B-series, defined by the Japanese Industrial Standards (JIS).

Practical Applications

• Business and Office:

  A4 is the standard for letters, reports, and office printing.

• Creative Arts:

  Origami enthusiasts refer to A-series sheets as "silver rectangles" due to their geometric properties.

• Envelopes:

  The C-series is derived to fit A-series sheets. For instance, a C4 envelope accommodates an unfolded A4 sheet.