Understanding Paper Size Ratios: A4 vs A3

Paper sizes in the A series, defined by the international standard ISO 216, follow a specific and consistent pattern. This standard ensures that each size maintains a specific aspect ratio and that larger sizes can be easily divided to create smaller ones.

The ISO 216 Standard and Aspect Ratio

The key principle behind the ISO 216 A series is the aspect ratio of \(1 : \sqrt{2}\). This means that if the shorter side of a sheet is \(w\) and the longer side is \(l\), then \(l = \sqrt{2} \times w\). This specific ratio has a useful property: if you cut a sheet of paper in half along its longer side, the resulting two smaller sheets will have the same \(1 : \sqrt{2}\) aspect ratio as the original sheet, but their area will be exactly half.

Each paper size in the series is obtained by cutting the previous size in half along its longer dimension. For example, an A1 sheet is obtained by cutting an A0 sheet in half, an A2 sheet is obtained by cutting an A1 sheet in half, and so on.

Dimensions of A4 and A3 Paper

Let's look at the standard dimensions for A4 and A3 paper sizes:

From the table, we can identify the required dimensions:

• The shorter side of an A4 sheet is 210 mm.
• The longer side of an A3 sheet is 420 mm.

Calculating the Ratio

We need to find the ratio of the shorter side of an A4 sheet to the longer side of an A3 sheet.

Ratio = (Shorter side of A4) : (Longer side of A3)

Ratio = \(210 \, mm : 420 \, mm\)

To simplify the ratio, divide both sides by the greatest common divisor, which is 210:

Ratio = \( \frac{210}{210} : \frac{420}{210} \)

Ratio = \(1 : 2\)

Thus, the ratio of the shorter side of an A4 sheet to the longer side of an A3 sheet is \(1 : 2\).

Summary of the Ratio Calculation

• A4 Shorter Side: 210 mm
• A3 Longer Side: 420 mm
• Ratio = \(210 : 420\)
• Simplify by dividing by 210: \(1 : 2\)

This result shows a clear relationship between the dimensions of A4 and A3 sizes based on the ISO 216 standard.

Revision Table: Paper Size Standards

Additional Information: ISO 216 Paper Sizes Explained

The ISO 216 standard is widely used globally and is based on the German DIN 476 standard. The brilliance of this system lies in its mathematical consistency, derived from the \(1:\sqrt{2}\) aspect ratio.

• Relationship Between Sizes: Cutting an An sheet along its longer side produces two An+1 sheets. This means the area of An+1 is exactly half the area of An. For instance, A4 is half the area of A3, A3 is half of A2, and so on.
• Mathematical Basis: If a rectangle has sides \(w\) and \(l\) with \(l/w = \sqrt{2}\), cutting it into two equal halves along the longer side results in two smaller rectangles with dimensions \(w\) and \(l/2\). The new aspect ratio is \(w / (l/2) = 2w/l = 2/\sqrt{2} = \sqrt{2}\). The ratio is preserved!
• Practical Use: This system is convenient because scaling documents is simple. For example, scaling a document from A4 to A3 is a simple enlargement, and two A4 pages fit perfectly onto one A3 page.
• A0 Area: The base size, A0, is defined to have an area of \(1 \, m^2\). With the aspect ratio \(l_0/w_0 = \sqrt{2}\) and area \(l_0 w_0 = 1 \, m^2\), you can derive the exact dimensions of A0 and all subsequent sizes.

Understanding this standard helps grasp why the ratio calculation between A4 and A3 dimensions results in a simple fraction.