Free Printable Dot Grid Paper for Bullet Journal - September Leather - Free Printable

Problem Analysis:

The image shows a series of nested rectangles labeled as A4, A5, and A6. These labels correspond to standard paper sizes in the ISO A-series, which follow specific mathematical relationships. The task is likely to determine the dimensions or ratios of these paper sizes based on their properties.

#### Key Properties of ISO A-Series Paper:
1. Area Relationship: Each subsequent size (e.g., A5 from A4) has exactly half the area of the previous size.
2. Shape Ratio: All sizes maintain the same aspect ratio, which is the square root of 2 (\(\sqrt{2}\)). This means the ratio of the longer side to the shorter side is always \(\sqrt{2}\).

Step-by-Step Solution:

#### 1. Understanding the Aspect Ratio
The aspect ratio for all A-series papers is \(\sqrt{2}\). If the shorter side of a paper is \(x\), then the longer side is \(x\sqrt{2}\).

#### 2. Standard Dimensions of A4 Paper
The standard dimensions of A4 paper are:
- Shorter side: \(210 \, \text{mm}\)
- Longer side: \(297 \, \text{mm}\)

We can verify the aspect ratio:
\[
\text{Aspect ratio} = \frac{\text{Longer side}}{\text{Shorter side}} = \frac{297}{210} \approx 1.414 \quad (\text{which is } \sqrt{2})
\]

#### 3. Deriving A5 from A4
A5 is obtained by folding A4 in half along its longer side. This halves the longer side of A4 while keeping the shorter side the same. Therefore:
- Shorter side of A5: \(210 \, \text{mm}\) (same as A4's shorter side)
- Longer side of A5: \(\frac{297}{2} = 148.5 \, \text{mm}\)

#### 4. Deriving A6 from A5
A6 is obtained by folding A5 in half along its longer side. This halves the longer side of A5 while keeping the shorter side the same. Therefore:
- Shorter side of A6: \(148.5 \, \text{mm}\) (same as A5's shorter side after folding)
- Longer side of A6: \(\frac{210}{2} = 105 \, \text{mm}\)

#### 5. Verifying the Aspect Ratio for A5 and A6
- For A5: \(\text{Aspect ratio} = \frac{210}{148.5} \approx 1.414\) (maintains \(\sqrt{2}\))
- For A6: \(\text{Aspect ratio} = \frac{148.5}{105} \approx 1.414\) (maintains \(\sqrt{2}\))

Final Answer:

The dimensions of the paper sizes are:
- A4: \(210 \, \text{mm} \times 297 \, \text{mm}\)
- A5: \(148.5 \, \text{mm} \times 210 \, \text{mm}\)
- A6: \(105 \, \text{mm} \times 148.5 \, \text{mm}\)

\[
\boxed{210 \, \text{mm} \times 297 \, \text{mm}, \, 148.5 \, \text{mm} \times 210 \, \text{mm}, \, 105 \, \text{mm} \times 148.5 \, \text{mm}}
\]