Pack the circles with radius equal to 14 cm using square and hexagon

Packing circles inside polygons like squares or hexagons depends on the geometric layout and efficiency of space usage. Let’s examine both configurations.

### 1. **Packing Circles in a Square:**
In a square, the packing can be done in a grid layout, meaning circles are arranged in rows and columns. However, due to gaps between the circles, this is not the most space-efficient method.

- **Side length of the square**: Assume the side length of the square is \( L \).
- **Radius of the circle**: Each circle has a radius of 14 cm, meaning the diameter is \( 2 \times 14 = 28 \) cm.

In an optimal square packing:
- Each row can fit \( \frac{L}{28} \) circles, and
- Each column can also fit \( \frac{L}{28} \) circles.

Thus, the total number of circles \( N \) packed in a square is approximately \( N = \left( \frac{L}{28} \right)^2 \).

### 2. **Packing Circles in a Hexagon:**
Hexagonal packing, also known as the closest packing, is much more space-efficient compared to square packing. In this arrangement, each circle is surrounded by six others, which minimizes wasted space.

- The distance between the centers of adjacent circles is equal to the diameter, i.e., 28 cm.
- For a regular hexagon, the number of circles packed depends on the size of the hexagon.

For a given side length \( s \) of the hexagon:
- Each side of the hexagon can fit \( \frac{s}{28} \) circles, but the arrangement allows slightly tighter packing, so the efficiency is greater than in the square case.

### Comparison of Efficiency:
- **Hexagonal packing** has a packing density of approximately **90.69%**.
- **Square packing** has a packing density of approximately **78.54%**.

This shows that **hexagonal packing is more efficient** in terms of space utilization.

Would you like further calculations on the number of circles you can pack in a specific area or explore detailed geometry of packing?

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**Relative Questions**:
1. How do hexagonal and square packings compare in real-world applications?
2. What formula can be used to calculate the number of circles packed in a hexagon of side length \( s \)?
3. What are the limitations of hexagonal packing in practical scenarios?
4. How does the packing density affect material usage in industries?
5. Can packing patterns be generalized for other shapes, such as triangles or rectangles?

**Tip**: When dealing with packing problems, always consider the packing density to ensure the most efficient use of space. Hexagonal packing is often the best choice when optimizing area usage.

Pack the circles with radius equal to 14 cm using square and hexagon.

Explore the geometry of packing circles with a radius of 14 cm inside squares and hexagons. Learn the efficiency of square and hexagonal packing, and compare their space usage with relevant formulas and densities. Suitable for Grades 9-12 students.

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